Research ArticleStability of Switched Server Systems with Constraints onService-Time and Capacity of Buffers

Li Wang Zhonghe He and Chi Zhang

Beijing Key Lab of Urban Intelligent Traffic Control Technology North China University of Technology Beijing 100144 China

Correspondence should be addressed to Zhonghe He zhonghehencuteducn

Received 6 February 2015 Revised 18 May 2015 Accepted 20 May 2015

Academic Editor Sebastian Anita

Copyright copy 2015 Li Wang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The execution of emptying policy ensures the convergence of any solution to the system to a unique periodic orbit which does notimpose constraints on service-time and capacity of buffers Motivated by these problems in this paper the service-time-limitedpolicy is first proposed based on the information resulted from the periodic orbit under emptying policy which imposes lowerand upper bounds on emptying time for the queue in each buffer by introducing lower-limit and upper-limit service-time factorsFurthermore the execution of service-time-limited policy in the case of finite buffer capacity is considered Moreover the notionof feasibility of states under service-time-limited policy is introduced and then the checking condition for feasibility of states isgiven that is the solution does not exceed the buffer capacity within the first cycle of the server At last a sufficient condition fordetermining upper-limit service-time factors ensuring that the given state is feasible is given

1 Introduction

Switched server system is a class of mathematical models forqueuing systems with finite number of conflicting queuesalternately served by a single server Moreover there exists anonzero setup time of the serverwhenever the server switchesfrom serving one queue to another one and assume thatthe jobs arrive at and leave the buffer at constant rates inthis paper The evolution of the system involves continuouschanges of queues in buffers and discrete switching of theserver and thus switched server system is a special class ofhybrid systems [1 2] with extensive applications in practicalproblems such as manufacturing systems [3 4] and trafficsignal control systems [5ndash7] and more applications of thisfield can be referred to [8]

Fundamental synthesis problem for switched server sys-tems is to design the scheduling policy of the server Theemptying policy (ie the server alternately empties queuesin buffers with any fixed cyclic sequence) was proposed in[9] under which any solution to the system asymptoticallyconverges to a unique periodic orbit analytically determinedby system parameters [6] However the emptying policydoes not impose constraints on queue-emptying time in

converging process of the solution In practical applicationsthe server with emptying policy must take longer time toempty buffers with larger queues and thus other buffers haveto wait longer time for service Thus in order to ensurefairness for all buffers the upper bound for emptying time ofeach buffer based on emptying policy was considered in [10]and a conjecture about stability of the policy was given whichwas further proved in [11] Also [12] considered distributedexecution of emptying policy with upper bounds for queue-emptying time of buffers in the networkwithmultiple serversIn most of literatures a scheduling policy is first proposedand then dynamic behaviors of the system are analyzed asin [9] In [13ndash16] a different idea for controlling the networkwas presented that is the steady state (a periodic orbit) ofthe system is first given and then corresponding schedulingpolicy is derived ensuring the convergence of any solutionto the steady state However the policies in [13ndash16] resultingfrom the given periodic orbit do not impose constraints onservice time of buffers

The problems about designs of the scheduling policieswith constraints on queue serving process mainly result frompractical applications For example in traffic intersection thesignal control for signalized intersections was modeled as

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 347931 10 pageshttpdxdoiorg1011552015347931

2 Mathematical Problems in Engineering

switched server systems in [5 6] and emptying policy wasapplied where signal light in a signalized intersection is seenas the server incoming links to the signalized intersection areseen as buffers which can accommodate queues of vehiclesthe lost time between phase switching is seen as the nonzerosetup time of the server and signal control law is seenas the scheduling policy of the server However in trafficcontrol [17] the shortest and longest green-time constraintson each of traffic phases are necessarily imposed for feasiblesignal control plans with the purpose of ensuring trafficsafety for drivers and pedestrians and controlling total delayof signalized intersections respectively Thus inspired bytraffic control the emptying policy is further extended inthis paper based on which the service-time-limited policy isproposed with lower and upper bounds on queue-emptyingtime of each buffer by introducing lower-limit and upper-limit service-time factors respectively Furthermore thebuffer capacity is finite for most of real-world problems Forexample in a signalized intersection incoming links withfinite length only accommodates finite number of vehiclesThus the execution of service-time-limited policy in thecase of finite buffer capacities is considered and moreoverthe notion of feasibility of states under service-time-limitedpolicy is introduced that is the state originating in whichthe solution asymptotically converges to the steady state (theperiodic orbit) and does not exceed buffer capacities in theconverging process Moreover the checking condition forfeasibility of states is given that is the solution does notexceed buffer capacitieswithin the first cycle of the server anda sufficient condition for determining upper-limit service-time factors ensuring that the given state is feasible is given

The paper is organized as follows After descriptions forthe model of switched server systems in Section 2 we intro-duce emptying and service-time-limited policies in Section 3Feasibility of states and checking conditions under service-time-limited policy are considered in Section 4 Conclusionsand future research topics are given in Section 5

2 Descriptions of Switched Server Systems

A switched server system (see Figure 1 for illustration) con-sists of 119899 (119899 ge 2) buffers and a single server where theserver alternately serves buffers in terms of the schedulingpolicy and only one buffer each time Let 119909

119894(119905) ge 0 denote

the queue of jobs in the buffer 119894 at the moment 119905 ge 0Because of nonnegative constraints on the queue of jobs ineach buffer the state space 119883 of the system is defined as119883 ≜ [1199091 119909119899]

119879isin R119899 119909

119894ge 0 119894 = 1 119899 Assume

that the jobs arrive at the buffer 119894 at a constant rate 119902119894gt 0

[lotss] Whenever the buffer 119894 in which there are accumu-lative queues that is 119909

119894(119905) gt 0 is served by the server the

jobs leave the buffer 119894 at a constant rate 119904119894gt 0 [lotss] and

whenever the buffer 119894 in which there are no accumulativequeues that is 119909

119894(119905) = 0 is served by the server the jobs

leave the buffer 119894 at the constant rate 119902119894 Both 119902

119894and 119904119894are

called arriving rate and service rate of jobs in the buffer119894 respectively and 119902

119894119904minus1119894

is called the load of the buffer 119894Whenever the server switches from serving the buffer 119894 to

Server

q1 qi

xix1 xn

qn

Buffer iBuffer 1 Buffer n

middot middot middot middot middot middot

si i = 1 n

Figure 1 A switched server system with 119899 buffers

the buffer 119895 there exists a nonzero setup time 119897119894119895gt 0 119894 119895 =

1 119899 119894 = 119895 [s] during which the server is in idleIn terms of above descriptions for switched server sys-

tems the dynamics of the queues of jobs in buffers can bedescribed by the following

Whenever the buffer 119894 with 119909119894(119905) gt 0 is served by the

server

(119905) = 119902 minus 119904119894119890119894 (1)

Whenever the buffer 119894 with 119909119894(119905) = 0 is served by the server

(119905) = 119902 minus 119902119894119890119894 (2)

Whenever the server switches from serving one buffer toanother one

(119905) = 119902 (3)

where 119909(119905) = [1199091(119905) 119909119899(119905)]119879isin 119883 119902 = [1199021 119902119899]

119879 and119890119894isin R119899 is 119899-dimensional unit vector that is the 119894th element

of 119890119894equals one and other elements of 119890

119894are zero

In the subsequent parts we assume that the total load ofbuffers satisfies

119899

sum

119895=1119902119895119904minus1119895lt 1 (4)

Obviously there is no equilibrium in the system describedby (1) (2) and (3) and the periodic orbit depending on thescheduling policy is the steady-state of the system whichattracts other trajectories of the system It was proved in[15] that the inequality (4) is the sufficient and necessarycondition for the existence of stable scheduling policy for thesystem

3 Stability of Scheduling Policy

In this section stability analysis of two scheduling policiesthat is emptying and service-time-limited policies is pre-sented where the service-time-limited policy admits service-time constraints on buffers based on emptying policy

Mathematical Problems in Engineering 3

31 Emptying Policy The emptying policy is described asfollows

(1) The buffers are served by the server in terms of anycyclic sequence for example 1 rarr 2 rarr sdot sdot sdot rarr 119899 rarr

1(2) Whenever the server switches from serving the buffer

119894 to the buffer 119894 + 1 (119894 = 1 119899 minus 1) there exists anonzero setup time 119897

119894119894+1 gt 0 and whenever the serverswitches from serving the buffer 119899 to the buffer 1 thesetup time is 119897

1198991 gt 0(3) When the buffer 119894 isin 1 119899 is being served the

service-time 119892119894(119896) for the queue is given by

119892119894 (119896) =

119909119894(119905119894

119896)

119904119894minus 119902119894

(5)

where 119905119894119896 119896 = 1 2 denotes the moment the server starts

serving the buffer 119894 within the 119896th cycle of the server 119909119894(119905119894

119896)

denotes the queue of jobs in the buffer 119894 at the moment 119905119894119896

and then 119909119894(119905119894

119896)(119904119894minus 119902119894)minus1 is the service-time for emptying the

queue 119909119894(119905119894

119896) in the buffer 119894

From the statements in emptying policy the server withnonzero setup times empties queues in buffers in terms ofcyclic sequence The following results hold

Theorem 1 (see [6]) Consider the switched server systemdescribed by (1) (2) and (3) under emptying policy Assumethat the total load of buffers satisfies (4) Then the followingstatements hold

(1) There exists a unique periodic orbit 119909119901(119905) = [1199091199011 (119905) 119909119901

119899(119905)]119879 to the system which is globally asymptotically

stable with respect to the state space119883(2) The period 119862 of the periodic orbit 119909119901(119905) is given by

119862 =

119871

1 minus sum119899119895=1 119902119895119904

minus1119895

(6)

where 119871 ≜ 11989712 + sdot sdot sdot + 119897119899minus1119899 + 1198971198991 is the total idle timewithin one cycle of the server

(3) For the periodic orbit 119909119901(119905) the service-time 119892119894for the

queue in the buffer 119894 is given by

119892119894= 119902119894119904minus1119894119862 (7)

Remark 2 The periodic orbit in Theorem 1 is denoted by119909119901(119905) in the succeeding parts It is derived from (6) and (7)

in Theorem 1 that the periodic orbit 119909119901(119905) can be uniquelydetermined by given system parameters and satisfy 119862 =

sum119899

119895=1 119892119895 + 119871 Importantly from (7) in Theorem 1 the signif-icance of the periodic orbit 119909119901(119905) is that within the period119862 and the total number of jobs arriving at the system isexactly equal to the total number of jobs leaving the systemat service rates Specifically if the signalized intersectionis modeled as a switched server system inequality (4) is

the undersaturated condition for signalized intersections andthe period 119862 is the minimum signal cycle (refer to detaileddiscussions in [6]) Moreover the consensus problems (iestates of the system can converge to a common value bylocal protocol) have become fundamental investigations incoordinated control of multiagent systems due to extensiveapplications in engineering fields (eg refer to [18 19])In the sense of traffic control the saturation level of somedirection is defined as the ratio of total number of vehiclesarriving at and leaving the intersection From the significanceof the periodic orbit 119909119901(119905) saturation levels are equal indifferent directions Then the emptying policy can realizethe consensus of saturation levels in traffic control implyingthe balance of traffic loads in different directions Thus theperiodic orbit 119909119901(119905) has practical meanings in applications totraffic control

32 Service-Time-Limited Policy The emptying policy doesnot restrict service-time for buffers However the problemof constraints on service-time of buffers is of importancein practical applications as stated in Introduction In thissubsection the service-time-limited policy is presented basedon emptying policy which can be described by the following

The first two terms (1) and (2) are the same as those indescriptions of emptying policy and (3) in emptying policyis replaced by the following

(31015840) When the buffer 119894 isin 1 119899 is being served theservice-time 119892

119894(119896) for the queue is given by

119892119894 (119896) =

119892119894 If

119909119894(119905119894

119896)

119904119894minus 119902119894

lt 119892119894

119909119894(119905119894

119896)

119904119894minus 119902119894

If 119892119894le

119909119894(119905119894

119896)

119904119894minus 119902119894

le 119892119894

119892119894 If

119909119894(119905119894

119896)

119904119894minus 119902119894

gt 119892119894

(8)

where 119892119894≜ 119892119894minus 119902119894119904minus1119894Γ119894min and 119892

119894≜ 119892119894+ 119902119894119904minus1119894Γ119894max

are respectively the shortest and longest service-timeassigned to the buffer 119894 where 119892

119894is given by (7) and

both Γ119894min and Γ

119894max are respectively called service-time lower-limit and upper-limit factors satisfying0 lt Γ119894min lt 119862 and Γ

119894max gt 0

The information resulted from 119862 and 119892119894of the periodic

orbit 119909119901(119905) determined inTheorem 1 is utilized for the designof service-time-limited policy From (8) the service-time119892119894(119896) of the buffer 119894 within the 119896th cycle is respectively

restricted by the shortest service-time 119892119894and longest service-

time 119892119894 If the queue-emptying time 119909

119894(119905119894

119896)(119904119894minus 119902119894)minus1 is less

than 119892119894assigned to the buffer 119894 then 119892

119894(119896) = 119892

119894 In this case

the serving process of the buffer 119894 is as follows the queuein the buffer 119894 is first served at the service-rate 119904

119894until the

queue is emptied (refer to dynamics in (1)) and then the buffer119894 is served at the arriving-rate 119902

119894until the shortest service-

time 119892119894ends (refer to dynamics in (2)) If the queue-emptying

time 119909119894(119905119894

119896)(119904119894minus 119902119894)minus1 is more than 119892

119894assigned to the buffer 119894

4 Mathematical Problems in Engineering

then 119892119894(119896) = 119892

119894 Otherwise the queue 119909

119894(119905119894

119896) in the buffer 119894 is

emptied and the server switches to the next bufferConsider the following inequality

119899

sum

119895=1119902119895119904minus1119895

lt minmin119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(9)

If Γ119894min and Γ

119894max satisfy 0 lt Γ1min = sdot sdot sdot = Γ119899min lt 119862 and

Γ1max = sdot sdot sdot = Γ119899max gt 0 then (9) is the same as (4)The following results hold for switched server systems

under service-time-limited policy

Theorem 3 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy Assume thatthe total load of buffers satisfies (9) Then any solution to thesystem asymptotically converges to the periodic orbit 119909119901(119905)

The proof of Theorem 3 can be referred to the appendixFurthermore consider the following two special cases forservice-time-limited policy

(C1) Γ1min = sdot sdot sdot = Γ119899min = 0 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 119892119894 119892119894gt 119892119894 119894 = 1 119899

(C2) Γ1min = sdot sdot sdot = Γ119899min = 119862 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 0 119892

119894gt 119892119894 119894 = 1 119899

Consider the following inequality

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(10)

If Γ1max = sdot sdot sdot = Γ119899max gt 0 is satisfied then (10) is the same as(4)

Theorem 4 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with factorssatisfying (C1) or (C2) Assume that the total load of bufferssatisfies (10) Then any solution to the system asymptoticallyconverges to the periodic orbit 119909119901(119905)

Proof When applying service-time-limited policy with fac-tors satisfying (C1) the statements in Theorem 4 can bederived by setting Γ

119894min = 0 119894 = 1 119899 in the proof ofTheorem 3 and when applying service-time-limited policywith factors satisfying (C2) the statements inTheorem 4 canbe derived by Cases 1 and 3 in the proof of Theorem 3 Inabove two cases (9) in the proof of Theorem 3 is changed to(10)

4 Feasibility of Service-Time-Limited Policy

Based on emptying policy service-time-limited policy admitsservice-time constraints on buffers by introducing service-time lower-limit and upper-limit factors Γ

119894min and Γ119894max 119894 =1 119899 but does not bring constraints on the buffer capacity

However the buffer capacity is finite for most of practicalproblems Thus we furthermore consider the execution ofservice-time-limited policy in case of finite buffer capacity

Let 119909max119894

gt 0 119894 = 1 119899 be the capacity of the buffer 119894defined as the maximum queue of jobs that the buffer 119894 canaccommodate Then the admissible region 119872 sub 119883 of thesystem is denoted as119872 ≜ [0 119909max

1 ] times sdot sdot sdot times [0 119909max119899

]It is derived from the significance of the periodic orbit

119909119901(119905) that the maximum queue of jobs in the buffer 119894 is given

by 119902119894(119862 minus 119892

119894) within the period 119862 Assume that the periodic

orbit 119909119901(119905) lies inside the admissible region119872 that is

119909max119894

gt 119902119894(119862minus119892

119894) 119894 = 1 119899 (11)

Definition 5 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policyThe state1199090 isin 119872 is called feasible if for the given service-time lower-limit factors Γ119891

119894min 119894 = 1 119899 there exist service-time upper-limit factors Γ119888

119894max gt 0 119894 = 1 119899 such that the solution119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in 1199090 asymptoticallyconverges to the periodic orbit 119909119901(119905) and moreover satisfies119909(119905) isin 119872 forall119905 ge 0

Furthermore it is deduced from (11) that there must existservice-time upper-limit factors Γ

119894max gt 0 119894 = 1 119899 satis-fying the following inequalities

119909max119894

ge 119902119894[

[

(119862minus119892119894) +sum

119895 =119894

119902119895119904minus1119895Γ119895max]

]

= 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119894 = 1 119899

(12)

Inequalities (12) indicate that when the queue of jobs in thebuffer 119894 is emptied the queue of jobs in the buffer 119894 does notexceed the buffer capacity after one cycle of the server Thefactors Γ

119894max 119894 = 1 119899 satisfying (12) are noted as Γ119888119894max 119894 =

1 119899 in the following parts

Theorem 6 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min and Γ119888

119894max gt 0 119894 = 1 119899 Assume that the total load ofbuffers satisfies (9) (or (10) if all Γ119891

119894min 119894 = 1 119899 satisfy (C1)or (C2)) and the state 1199090 isin 119872 has the property that thesolution 119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in the state 1199090satisfies the condition [1199091(11990511) 119909119899(119905

119899

1)]119879isin 119872 where 1199051198941 is

the moment the server starts serving the buffer 119894 within the firstcycle of the serverThen the state1199090 isin 119872 is feasible with respectto Γ119888119894max gt 0 119894 = 1 119899

Theproof ofTheorem 6 can be referred to the appendix Itis derived from Theorem 6 that the checking conditionfor feasibility of the state is that the corresponding solutiondoes not exceed the buffer capacity within the first cycleof the server with given Γ

119891

119894min and Γ119888

119894max 119894 = 1 119899Accordingly the feasible region 1198830[Γ

119891

119894min Γ119888

119894max] sube 119872 thatis all of feasible states with respect to Γ

119891

119894min and Γ119888

119894max 119894 =

Mathematical Problems in Engineering 5

1 119899 can be obtained from the checking condition forfeasibility of the state Specifically analytic expression offeasible region 1198830[Γ

119891

119894min Γ119888

119894max] for switched server systemswith two buffers can be easily determined as follows

(1) If 0 le Γ119891

119894min lt 119862 119894 = 1 2 then 1198830[Γ119891

119894min Γ119888

119894max] =

11988310 cup 119883

20 cup 119883

30 where 119883

119894

0 119894 = 1 2 3 are respectivelygiven by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1lt 1198921 minus 1199021119904

minus11 Γ119891

1min

1199092 (0) + 1199022 [(1198921 minus 1199021119904minus11 Γ119891

1min) + 11989712] le 119909max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198921 minus 1199021119904minus11 Γ119891

1min le 1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988330 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(13)

(2) If Γ119891119894min = 119862 119894 = 1 2 then1198830[Γ

119891

119894min Γ119888

119894max] = 11988310 cup119883

20

where1198831198940 119894 = 1 2 are respectively given by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(14)

From Theorem 6 feasibility of the state depends onchoices of factors Γ119891

119894min and Γ119888

119894max 119894 = 1 119899 Howeverinfeasibility of the state with respect to some given factorsΓ119891

119894min and Γ119888119894max 119894 = 1 119899 does not imply inexistence of

factors ensuring the state is feasible Furthermore we con-sider the problem of how to solve factors Γ119888

119894max 119894 = 1 119899such that the given state is feasible with given Γ

119891

119894min 119894 =

1 119899If service-time-limited policy is applied with given

Γ119891

119894min = 0 or 0 lt Γ119891119894min lt 119862 119894 = 1 119899 in terms of the check-

ing condition for feasibility of states in Theorem 6 the givenstate 1199090 = [1199091(0) 119909119899(0)]

119879isin 119872 is infeasible if at least one

of the following inequalities holds

119909119894 (0) gt 119909max

119894

minus 119902119894[

[

119894minus1sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) +

119894minus1sum

119895=1119897119895119895+1]

]

119894 = 2 119899

(15)

The significance of (15) is that the queue in the buffer 119894 ge 2exceeds the buffer capacity even if all of buffers 119895 119895 =

1 119894minus1 are servedwith the shortest service-timewithin thefirst cycle of the server Furthermore a sufficient condition isgiven for determining Γ119888

119894max 119894 = 1 119899 ensuring the givenstate 1199090 isin 119872 is feasible

Proposition 7 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min 119894 = 1 119899 For the given state 1199090 isin 119872 if the set Ω(1199090)of factors [Γ1max Γ119899max]

119879 given by (16) is nonempty andthere exists [Γ1198881max Γ

119888

119899max]119879isin Ω(1199090) such that the total

load of buffers satisfies (9) (or (10) if all Γ119891119894min 119894 = 1 119899

satisfy (C1) or (C2)) then the given state 1199090 isin 119872 is feasiblewith respect to [Γ1198881max Γ

119888

119899max]119879

(a)

sum

119895 =1119902119895119904minus1119895Γ119895max le

119909max11199021

minus sum

119895 =1119892119895minus 119871

sum

119895 =119899

119902119895119904minus1119895Γ119895max le

119909max119899

119902119899

minus sum

119895 =119899

119892119895minus 119871

(b)

1199021119904minus11 Γ1max le

[119909max2 minus 1199092 (0)]

1199022minus 1198921 minus 11989712

119899minus1sum

119895=1119902119895119904minus1119895Γ119895max le

[119909max119899

minus 119909119899 (0)]

119902119899

minus

119899minus1sum

119895=1119892119895minus

119899minus1sum

119895=1119897119895119895+1

(c)

0 lt Γ1max le Γlowast

1max

0 lt Γ119899max le Γ

lowast

119899max

(16)

Proof Inequalities (a) in (16) imply that (12) holds and wecan derive from (b) in (16) that

1199092 (11990521) le 1199092 (0) + 1199022 [(1198921 + 1199021119904

minus11 Γ1max) + 11989712]

le 119909max2

119909119899(119905119899

1) le 119909119899 (0)

+ 119902119899[

[

119899minus1sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) +

119899minus1sum

119895=1119897119895119895+1]

]

le 119909max119899

(17)

6 Mathematical Problems in Engineering

which indicates that the solution does not exceed the buffercapacity within the first cycle of the server FurthermoreΓlowast

119894max 119894 = 1 119899 in (c) are the maximum allowable service-time upper-limit factors Thus from Theorem 6 the givenstate 1199090 isin 119872 is feasible

5 Conclusions

For most of real-world problems about queuing systemsservice-times and queues of buffers must be constrained Inthis paper inspired by practical problems in traffic controlthe service-time-limited policy is proposed which is theextension to emptying policy Moreover the execution ofservice-time-limited policy in the case of finite buffer capaci-ties is considered and the notion of feasibility of states underservice-time-limited policy is presented Furthermore basedon the checking condition for feasibility of states (ie thesolution does not exceed buffer capacities within the firstcycle of the server) a sufficient condition for determiningfeasibility of states is given

The scheduling policy proposed in this paper admitstaking into consideration service-time and queue constraintson buffers by the introduction of the notion of feasibility ofstates and service-time upper-limit factors for the feasiblestate can be solved by testing the nonempty set Ω(1199090) Thusour results can be applied to traffic control as stated inthe Introduction especially in critical saturation case forexample the length of queues of vehicles on incoming linksmay be larger with lower traffic loads satisfying (4) Signalcontrol of T-shape intersection is typical application of ourresults which can be referred to [6] for details

From views of traffic control the server may servemultiple nonconflicting flows which is our further researchextension of results in the paper

Appendix

Proof of Theorem 3 Assume that 119905119894119896and 119879

119894

119896 respectively

represent moments that the server starts and finishes servingthe queue in the buffer 119894 in terms of service-time-limitedpolicy within the 119896th cycle of the server forall119894 isin 1 119899 119896 =1 2 Then 1199051198941 is the moment that the server starts servingthe buffer 119894 within the first cycle of the server Considerthe following three possible cases for any solution 119909(119905) =

[1199091(119905) 119909119899(119905)]119879 to the system originating in the initial state

1199090 isin 119883

Case 1 119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894(119905

119894

1)(119904119894 minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max

Case 2 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min

Case3 119909119894(119905119894

1)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1

119894Γ119894max

We prove that the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879

asymptotically converges to the periodic orbit 119909119901(119905) in anycase above

Case 1 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1) (119904119894 minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A1)

then the queue-emptying time 119909119894(119905119894

119898)(119904119894minus 119902119894)minus1 forall119898 ge 1 of

the buffer 119894 within any cycle satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A2)

Proof of Case 1 We prove Case 1 by using mathematicalinduction From (A1) Case 1 holds with119898 = 1 Furthermoreassume that Case 1 holds with some 119898 ge 1 then in terms ofservice-time-limited policy we have that 119909

119894(119879119894

119898) = 0 and

119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

le 119909119894(119905119894

119898+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

(A3)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895minus119902119895119904minus1119895Γ119895min)+119871]

satisfies

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] minus 119902119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

minus

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895min

1 minus 119902119894119904minus1119894

ge 119892119894minus

119902119894119904minus1119894max119895isin1119899 Γ119895minsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A4)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

(A5)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le [1minus 119902119894119904minus1119894]

Γ119894min

max119895isin1119899 Γ119895min

(A6)

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

switched server systems in [5 6] and emptying policy wasapplied where signal light in a signalized intersection is seenas the server incoming links to the signalized intersection areseen as buffers which can accommodate queues of vehiclesthe lost time between phase switching is seen as the nonzerosetup time of the server and signal control law is seenas the scheduling policy of the server However in trafficcontrol [17] the shortest and longest green-time constraintson each of traffic phases are necessarily imposed for feasiblesignal control plans with the purpose of ensuring trafficsafety for drivers and pedestrians and controlling total delayof signalized intersections respectively Thus inspired bytraffic control the emptying policy is further extended inthis paper based on which the service-time-limited policy isproposed with lower and upper bounds on queue-emptyingtime of each buffer by introducing lower-limit and upper-limit service-time factors respectively Furthermore thebuffer capacity is finite for most of real-world problems Forexample in a signalized intersection incoming links withfinite length only accommodates finite number of vehiclesThus the execution of service-time-limited policy in thecase of finite buffer capacities is considered and moreoverthe notion of feasibility of states under service-time-limitedpolicy is introduced that is the state originating in whichthe solution asymptotically converges to the steady state (theperiodic orbit) and does not exceed buffer capacities in theconverging process Moreover the checking condition forfeasibility of states is given that is the solution does notexceed buffer capacitieswithin the first cycle of the server anda sufficient condition for determining upper-limit service-time factors ensuring that the given state is feasible is given

The paper is organized as follows After descriptions forthe model of switched server systems in Section 2 we intro-duce emptying and service-time-limited policies in Section 3Feasibility of states and checking conditions under service-time-limited policy are considered in Section 4 Conclusionsand future research topics are given in Section 5

2 Descriptions of Switched Server Systems

A switched server system (see Figure 1 for illustration) con-sists of 119899 (119899 ge 2) buffers and a single server where theserver alternately serves buffers in terms of the schedulingpolicy and only one buffer each time Let 119909

119894(119905) ge 0 denote

the queue of jobs in the buffer 119894 at the moment 119905 ge 0Because of nonnegative constraints on the queue of jobs ineach buffer the state space 119883 of the system is defined as119883 ≜ [1199091 119909119899]

119879isin R119899 119909

119894ge 0 119894 = 1 119899 Assume

that the jobs arrive at the buffer 119894 at a constant rate 119902119894gt 0

[lotss] Whenever the buffer 119894 in which there are accumu-lative queues that is 119909

119894(119905) gt 0 is served by the server the

jobs leave the buffer 119894 at a constant rate 119904119894gt 0 [lotss] and

whenever the buffer 119894 in which there are no accumulativequeues that is 119909

119894(119905) = 0 is served by the server the jobs

leave the buffer 119894 at the constant rate 119902119894 Both 119902

119894and 119904119894are

called arriving rate and service rate of jobs in the buffer119894 respectively and 119902

119894119904minus1119894

is called the load of the buffer 119894Whenever the server switches from serving the buffer 119894 to

Server

q1 qi

xix1 xn

qn

Buffer iBuffer 1 Buffer n

middot middot middot middot middot middot

si i = 1 n

Figure 1 A switched server system with 119899 buffers

the buffer 119895 there exists a nonzero setup time 119897119894119895gt 0 119894 119895 =

1 119899 119894 = 119895 [s] during which the server is in idleIn terms of above descriptions for switched server sys-

tems the dynamics of the queues of jobs in buffers can bedescribed by the following

Whenever the buffer 119894 with 119909119894(119905) gt 0 is served by the

server

(119905) = 119902 minus 119904119894119890119894 (1)

Whenever the buffer 119894 with 119909119894(119905) = 0 is served by the server

(119905) = 119902 minus 119902119894119890119894 (2)

Whenever the server switches from serving one buffer toanother one

(119905) = 119902 (3)

where 119909(119905) = [1199091(119905) 119909119899(119905)]119879isin 119883 119902 = [1199021 119902119899]

119879 and119890119894isin R119899 is 119899-dimensional unit vector that is the 119894th element

of 119890119894equals one and other elements of 119890

119894are zero

In the subsequent parts we assume that the total load ofbuffers satisfies

119899

sum

119895=1119902119895119904minus1119895lt 1 (4)

Obviously there is no equilibrium in the system describedby (1) (2) and (3) and the periodic orbit depending on thescheduling policy is the steady-state of the system whichattracts other trajectories of the system It was proved in[15] that the inequality (4) is the sufficient and necessarycondition for the existence of stable scheduling policy for thesystem

3 Stability of Scheduling Policy

In this section stability analysis of two scheduling policiesthat is emptying and service-time-limited policies is pre-sented where the service-time-limited policy admits service-time constraints on buffers based on emptying policy

Mathematical Problems in Engineering 3

31 Emptying Policy The emptying policy is described asfollows

(1) The buffers are served by the server in terms of anycyclic sequence for example 1 rarr 2 rarr sdot sdot sdot rarr 119899 rarr

1(2) Whenever the server switches from serving the buffer

119894 to the buffer 119894 + 1 (119894 = 1 119899 minus 1) there exists anonzero setup time 119897

119894119894+1 gt 0 and whenever the serverswitches from serving the buffer 119899 to the buffer 1 thesetup time is 119897

1198991 gt 0(3) When the buffer 119894 isin 1 119899 is being served the

service-time 119892119894(119896) for the queue is given by

119892119894 (119896) =

119909119894(119905119894

119896)

119904119894minus 119902119894

(5)

where 119905119894119896 119896 = 1 2 denotes the moment the server starts

serving the buffer 119894 within the 119896th cycle of the server 119909119894(119905119894

119896)

denotes the queue of jobs in the buffer 119894 at the moment 119905119894119896

and then 119909119894(119905119894

119896)(119904119894minus 119902119894)minus1 is the service-time for emptying the

queue 119909119894(119905119894

119896) in the buffer 119894

From the statements in emptying policy the server withnonzero setup times empties queues in buffers in terms ofcyclic sequence The following results hold

Theorem 1 (see [6]) Consider the switched server systemdescribed by (1) (2) and (3) under emptying policy Assumethat the total load of buffers satisfies (4) Then the followingstatements hold

(1) There exists a unique periodic orbit 119909119901(119905) = [1199091199011 (119905) 119909119901

119899(119905)]119879 to the system which is globally asymptotically

stable with respect to the state space119883(2) The period 119862 of the periodic orbit 119909119901(119905) is given by

119862 =

119871

1 minus sum119899119895=1 119902119895119904

minus1119895

(6)

where 119871 ≜ 11989712 + sdot sdot sdot + 119897119899minus1119899 + 1198971198991 is the total idle timewithin one cycle of the server

(3) For the periodic orbit 119909119901(119905) the service-time 119892119894for the

queue in the buffer 119894 is given by

119892119894= 119902119894119904minus1119894119862 (7)

Remark 2 The periodic orbit in Theorem 1 is denoted by119909119901(119905) in the succeeding parts It is derived from (6) and (7)

in Theorem 1 that the periodic orbit 119909119901(119905) can be uniquelydetermined by given system parameters and satisfy 119862 =

sum119899

119895=1 119892119895 + 119871 Importantly from (7) in Theorem 1 the signif-icance of the periodic orbit 119909119901(119905) is that within the period119862 and the total number of jobs arriving at the system isexactly equal to the total number of jobs leaving the systemat service rates Specifically if the signalized intersectionis modeled as a switched server system inequality (4) is

the undersaturated condition for signalized intersections andthe period 119862 is the minimum signal cycle (refer to detaileddiscussions in [6]) Moreover the consensus problems (iestates of the system can converge to a common value bylocal protocol) have become fundamental investigations incoordinated control of multiagent systems due to extensiveapplications in engineering fields (eg refer to [18 19])In the sense of traffic control the saturation level of somedirection is defined as the ratio of total number of vehiclesarriving at and leaving the intersection From the significanceof the periodic orbit 119909119901(119905) saturation levels are equal indifferent directions Then the emptying policy can realizethe consensus of saturation levels in traffic control implyingthe balance of traffic loads in different directions Thus theperiodic orbit 119909119901(119905) has practical meanings in applications totraffic control

32 Service-Time-Limited Policy The emptying policy doesnot restrict service-time for buffers However the problemof constraints on service-time of buffers is of importancein practical applications as stated in Introduction In thissubsection the service-time-limited policy is presented basedon emptying policy which can be described by the following

The first two terms (1) and (2) are the same as those indescriptions of emptying policy and (3) in emptying policyis replaced by the following

(31015840) When the buffer 119894 isin 1 119899 is being served theservice-time 119892

119894(119896) for the queue is given by

119892119894 (119896) =

119892119894 If

119909119894(119905119894

119896)

119904119894minus 119902119894

lt 119892119894

119909119894(119905119894

119896)

119904119894minus 119902119894

If 119892119894le

119909119894(119905119894

119896)

119904119894minus 119902119894

le 119892119894

119892119894 If

119909119894(119905119894

119896)

119904119894minus 119902119894

gt 119892119894

(8)

where 119892119894≜ 119892119894minus 119902119894119904minus1119894Γ119894min and 119892

119894≜ 119892119894+ 119902119894119904minus1119894Γ119894max

are respectively the shortest and longest service-timeassigned to the buffer 119894 where 119892

119894is given by (7) and

both Γ119894min and Γ

119894max are respectively called service-time lower-limit and upper-limit factors satisfying0 lt Γ119894min lt 119862 and Γ

119894max gt 0

The information resulted from 119862 and 119892119894of the periodic

orbit 119909119901(119905) determined inTheorem 1 is utilized for the designof service-time-limited policy From (8) the service-time119892119894(119896) of the buffer 119894 within the 119896th cycle is respectively

restricted by the shortest service-time 119892119894and longest service-

time 119892119894 If the queue-emptying time 119909

119894(119905119894

119896)(119904119894minus 119902119894)minus1 is less

than 119892119894assigned to the buffer 119894 then 119892

119894(119896) = 119892

119894 In this case

the serving process of the buffer 119894 is as follows the queuein the buffer 119894 is first served at the service-rate 119904

119894until the

queue is emptied (refer to dynamics in (1)) and then the buffer119894 is served at the arriving-rate 119902

119894until the shortest service-

time 119892119894ends (refer to dynamics in (2)) If the queue-emptying

time 119909119894(119905119894

119896)(119904119894minus 119902119894)minus1 is more than 119892

119894assigned to the buffer 119894

4 Mathematical Problems in Engineering

then 119892119894(119896) = 119892

119894 Otherwise the queue 119909

119894(119905119894

119896) in the buffer 119894 is

emptied and the server switches to the next bufferConsider the following inequality

119899

sum

119895=1119902119895119904minus1119895

lt minmin119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(9)

If Γ119894min and Γ

119894max satisfy 0 lt Γ1min = sdot sdot sdot = Γ119899min lt 119862 and

Γ1max = sdot sdot sdot = Γ119899max gt 0 then (9) is the same as (4)The following results hold for switched server systems

under service-time-limited policy

Theorem 3 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy Assume thatthe total load of buffers satisfies (9) Then any solution to thesystem asymptotically converges to the periodic orbit 119909119901(119905)

The proof of Theorem 3 can be referred to the appendixFurthermore consider the following two special cases forservice-time-limited policy

(C1) Γ1min = sdot sdot sdot = Γ119899min = 0 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 119892119894 119892119894gt 119892119894 119894 = 1 119899

(C2) Γ1min = sdot sdot sdot = Γ119899min = 119862 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 0 119892

119894gt 119892119894 119894 = 1 119899

Consider the following inequality

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(10)

If Γ1max = sdot sdot sdot = Γ119899max gt 0 is satisfied then (10) is the same as(4)

Theorem 4 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with factorssatisfying (C1) or (C2) Assume that the total load of bufferssatisfies (10) Then any solution to the system asymptoticallyconverges to the periodic orbit 119909119901(119905)

Proof When applying service-time-limited policy with fac-tors satisfying (C1) the statements in Theorem 4 can bederived by setting Γ

119894min = 0 119894 = 1 119899 in the proof ofTheorem 3 and when applying service-time-limited policywith factors satisfying (C2) the statements inTheorem 4 canbe derived by Cases 1 and 3 in the proof of Theorem 3 Inabove two cases (9) in the proof of Theorem 3 is changed to(10)

4 Feasibility of Service-Time-Limited Policy

Based on emptying policy service-time-limited policy admitsservice-time constraints on buffers by introducing service-time lower-limit and upper-limit factors Γ

119894min and Γ119894max 119894 =1 119899 but does not bring constraints on the buffer capacity

However the buffer capacity is finite for most of practicalproblems Thus we furthermore consider the execution ofservice-time-limited policy in case of finite buffer capacity

Let 119909max119894

gt 0 119894 = 1 119899 be the capacity of the buffer 119894defined as the maximum queue of jobs that the buffer 119894 canaccommodate Then the admissible region 119872 sub 119883 of thesystem is denoted as119872 ≜ [0 119909max

1 ] times sdot sdot sdot times [0 119909max119899

]It is derived from the significance of the periodic orbit

119909119901(119905) that the maximum queue of jobs in the buffer 119894 is given

by 119902119894(119862 minus 119892

119894) within the period 119862 Assume that the periodic

orbit 119909119901(119905) lies inside the admissible region119872 that is

119909max119894

gt 119902119894(119862minus119892

119894) 119894 = 1 119899 (11)

Definition 5 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policyThe state1199090 isin 119872 is called feasible if for the given service-time lower-limit factors Γ119891

119894min 119894 = 1 119899 there exist service-time upper-limit factors Γ119888

119894max gt 0 119894 = 1 119899 such that the solution119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in 1199090 asymptoticallyconverges to the periodic orbit 119909119901(119905) and moreover satisfies119909(119905) isin 119872 forall119905 ge 0

Furthermore it is deduced from (11) that there must existservice-time upper-limit factors Γ

119894max gt 0 119894 = 1 119899 satis-fying the following inequalities

119909max119894

ge 119902119894[

[

(119862minus119892119894) +sum

119895 =119894

119902119895119904minus1119895Γ119895max]

]

= 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119894 = 1 119899

(12)

Inequalities (12) indicate that when the queue of jobs in thebuffer 119894 is emptied the queue of jobs in the buffer 119894 does notexceed the buffer capacity after one cycle of the server Thefactors Γ

119894max 119894 = 1 119899 satisfying (12) are noted as Γ119888119894max 119894 =

1 119899 in the following parts

Theorem 6 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min and Γ119888

119894max gt 0 119894 = 1 119899 Assume that the total load ofbuffers satisfies (9) (or (10) if all Γ119891

119894min 119894 = 1 119899 satisfy (C1)or (C2)) and the state 1199090 isin 119872 has the property that thesolution 119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in the state 1199090satisfies the condition [1199091(11990511) 119909119899(119905

119899

1)]119879isin 119872 where 1199051198941 is

the moment the server starts serving the buffer 119894 within the firstcycle of the serverThen the state1199090 isin 119872 is feasible with respectto Γ119888119894max gt 0 119894 = 1 119899

Theproof ofTheorem 6 can be referred to the appendix Itis derived from Theorem 6 that the checking conditionfor feasibility of the state is that the corresponding solutiondoes not exceed the buffer capacity within the first cycleof the server with given Γ

119891

119894min and Γ119888

119894max 119894 = 1 119899Accordingly the feasible region 1198830[Γ

119891

119894min Γ119888

119894max] sube 119872 thatis all of feasible states with respect to Γ

119891

119894min and Γ119888

119894max 119894 =

Mathematical Problems in Engineering 5

1 119899 can be obtained from the checking condition forfeasibility of the state Specifically analytic expression offeasible region 1198830[Γ

119891

119894min Γ119888

119894max] for switched server systemswith two buffers can be easily determined as follows

(1) If 0 le Γ119891

119894min lt 119862 119894 = 1 2 then 1198830[Γ119891

119894min Γ119888

119894max] =

11988310 cup 119883

20 cup 119883

30 where 119883

119894

0 119894 = 1 2 3 are respectivelygiven by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1lt 1198921 minus 1199021119904

minus11 Γ119891

1min

1199092 (0) + 1199022 [(1198921 minus 1199021119904minus11 Γ119891

1min) + 11989712] le 119909max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198921 minus 1199021119904minus11 Γ119891

1min le 1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988330 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(13)

(2) If Γ119891119894min = 119862 119894 = 1 2 then1198830[Γ

119891

119894min Γ119888

119894max] = 11988310 cup119883

20

where1198831198940 119894 = 1 2 are respectively given by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(14)

From Theorem 6 feasibility of the state depends onchoices of factors Γ119891

119894min and Γ119888

119894max 119894 = 1 119899 Howeverinfeasibility of the state with respect to some given factorsΓ119891

119894min and Γ119888119894max 119894 = 1 119899 does not imply inexistence of

factors ensuring the state is feasible Furthermore we con-sider the problem of how to solve factors Γ119888

119894max 119894 = 1 119899such that the given state is feasible with given Γ

119891

119894min 119894 =

1 119899If service-time-limited policy is applied with given

Γ119891

119894min = 0 or 0 lt Γ119891119894min lt 119862 119894 = 1 119899 in terms of the check-

ing condition for feasibility of states in Theorem 6 the givenstate 1199090 = [1199091(0) 119909119899(0)]

119879isin 119872 is infeasible if at least one

of the following inequalities holds

119909119894 (0) gt 119909max

119894

minus 119902119894[

[

119894minus1sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) +

119894minus1sum

119895=1119897119895119895+1]

]

119894 = 2 119899

(15)

The significance of (15) is that the queue in the buffer 119894 ge 2exceeds the buffer capacity even if all of buffers 119895 119895 =

1 119894minus1 are servedwith the shortest service-timewithin thefirst cycle of the server Furthermore a sufficient condition isgiven for determining Γ119888

119894max 119894 = 1 119899 ensuring the givenstate 1199090 isin 119872 is feasible

Proposition 7 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min 119894 = 1 119899 For the given state 1199090 isin 119872 if the set Ω(1199090)of factors [Γ1max Γ119899max]

119879 given by (16) is nonempty andthere exists [Γ1198881max Γ

119888

119899max]119879isin Ω(1199090) such that the total

load of buffers satisfies (9) (or (10) if all Γ119891119894min 119894 = 1 119899

satisfy (C1) or (C2)) then the given state 1199090 isin 119872 is feasiblewith respect to [Γ1198881max Γ

119888

119899max]119879

(a)

sum

119895 =1119902119895119904minus1119895Γ119895max le

119909max11199021

minus sum

119895 =1119892119895minus 119871

sum

119895 =119899

119902119895119904minus1119895Γ119895max le

119909max119899

119902119899

minus sum

119895 =119899

119892119895minus 119871

(b)

1199021119904minus11 Γ1max le

[119909max2 minus 1199092 (0)]

1199022minus 1198921 minus 11989712

119899minus1sum

119895=1119902119895119904minus1119895Γ119895max le

[119909max119899

minus 119909119899 (0)]

119902119899

minus

119899minus1sum

119895=1119892119895minus

119899minus1sum

119895=1119897119895119895+1

(c)

0 lt Γ1max le Γlowast

1max

0 lt Γ119899max le Γ

lowast

119899max

(16)

Proof Inequalities (a) in (16) imply that (12) holds and wecan derive from (b) in (16) that

1199092 (11990521) le 1199092 (0) + 1199022 [(1198921 + 1199021119904

minus11 Γ1max) + 11989712]

le 119909max2

119909119899(119905119899

1) le 119909119899 (0)

+ 119902119899[

[

119899minus1sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) +

119899minus1sum

119895=1119897119895119895+1]

]

le 119909max119899

(17)

6 Mathematical Problems in Engineering

which indicates that the solution does not exceed the buffercapacity within the first cycle of the server FurthermoreΓlowast

119894max 119894 = 1 119899 in (c) are the maximum allowable service-time upper-limit factors Thus from Theorem 6 the givenstate 1199090 isin 119872 is feasible

5 Conclusions

For most of real-world problems about queuing systemsservice-times and queues of buffers must be constrained Inthis paper inspired by practical problems in traffic controlthe service-time-limited policy is proposed which is theextension to emptying policy Moreover the execution ofservice-time-limited policy in the case of finite buffer capaci-ties is considered and the notion of feasibility of states underservice-time-limited policy is presented Furthermore basedon the checking condition for feasibility of states (ie thesolution does not exceed buffer capacities within the firstcycle of the server) a sufficient condition for determiningfeasibility of states is given

The scheduling policy proposed in this paper admitstaking into consideration service-time and queue constraintson buffers by the introduction of the notion of feasibility ofstates and service-time upper-limit factors for the feasiblestate can be solved by testing the nonempty set Ω(1199090) Thusour results can be applied to traffic control as stated inthe Introduction especially in critical saturation case forexample the length of queues of vehicles on incoming linksmay be larger with lower traffic loads satisfying (4) Signalcontrol of T-shape intersection is typical application of ourresults which can be referred to [6] for details

From views of traffic control the server may servemultiple nonconflicting flows which is our further researchextension of results in the paper

Appendix

Proof of Theorem 3 Assume that 119905119894119896and 119879

119894

119896 respectively

represent moments that the server starts and finishes servingthe queue in the buffer 119894 in terms of service-time-limitedpolicy within the 119896th cycle of the server forall119894 isin 1 119899 119896 =1 2 Then 1199051198941 is the moment that the server starts servingthe buffer 119894 within the first cycle of the server Considerthe following three possible cases for any solution 119909(119905) =

[1199091(119905) 119909119899(119905)]119879 to the system originating in the initial state

1199090 isin 119883

Case 1 119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894(119905

119894

1)(119904119894 minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max

Case 2 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min

Case3 119909119894(119905119894

1)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1

119894Γ119894max

We prove that the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879

asymptotically converges to the periodic orbit 119909119901(119905) in anycase above

Case 1 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1) (119904119894 minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A1)

then the queue-emptying time 119909119894(119905119894

119898)(119904119894minus 119902119894)minus1 forall119898 ge 1 of

the buffer 119894 within any cycle satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A2)

Proof of Case 1 We prove Case 1 by using mathematicalinduction From (A1) Case 1 holds with119898 = 1 Furthermoreassume that Case 1 holds with some 119898 ge 1 then in terms ofservice-time-limited policy we have that 119909

119894(119879119894

119898) = 0 and

119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

le 119909119894(119905119894

119898+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

(A3)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895minus119902119895119904minus1119895Γ119895min)+119871]

satisfies

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] minus 119902119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

minus

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895min

1 minus 119902119894119904minus1119894

ge 119892119894minus

119902119894119904minus1119894max119895isin1119899 Γ119895minsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A4)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

(A5)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le [1minus 119902119894119904minus1119894]

Γ119894min

max119895isin1119899 Γ119895min

(A6)

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

31 Emptying Policy The emptying policy is described asfollows

(1) The buffers are served by the server in terms of anycyclic sequence for example 1 rarr 2 rarr sdot sdot sdot rarr 119899 rarr

1(2) Whenever the server switches from serving the buffer

119894 to the buffer 119894 + 1 (119894 = 1 119899 minus 1) there exists anonzero setup time 119897

119894119894+1 gt 0 and whenever the serverswitches from serving the buffer 119899 to the buffer 1 thesetup time is 119897

1198991 gt 0(3) When the buffer 119894 isin 1 119899 is being served the

service-time 119892119894(119896) for the queue is given by

119892119894 (119896) =

119909119894(119905119894

119896)

119904119894minus 119902119894

(5)

where 119905119894119896 119896 = 1 2 denotes the moment the server starts

serving the buffer 119894 within the 119896th cycle of the server 119909119894(119905119894

119896)

denotes the queue of jobs in the buffer 119894 at the moment 119905119894119896

and then 119909119894(119905119894

119896)(119904119894minus 119902119894)minus1 is the service-time for emptying the

queue 119909119894(119905119894

119896) in the buffer 119894

From the statements in emptying policy the server withnonzero setup times empties queues in buffers in terms ofcyclic sequence The following results hold

Theorem 1 (see [6]) Consider the switched server systemdescribed by (1) (2) and (3) under emptying policy Assumethat the total load of buffers satisfies (4) Then the followingstatements hold

(1) There exists a unique periodic orbit 119909119901(119905) = [1199091199011 (119905) 119909119901

119899(119905)]119879 to the system which is globally asymptotically

stable with respect to the state space119883(2) The period 119862 of the periodic orbit 119909119901(119905) is given by

119862 =

119871

1 minus sum119899119895=1 119902119895119904

minus1119895

(6)

where 119871 ≜ 11989712 + sdot sdot sdot + 119897119899minus1119899 + 1198971198991 is the total idle timewithin one cycle of the server

(3) For the periodic orbit 119909119901(119905) the service-time 119892119894for the

queue in the buffer 119894 is given by

119892119894= 119902119894119904minus1119894119862 (7)

Remark 2 The periodic orbit in Theorem 1 is denoted by119909119901(119905) in the succeeding parts It is derived from (6) and (7)

in Theorem 1 that the periodic orbit 119909119901(119905) can be uniquelydetermined by given system parameters and satisfy 119862 =

sum119899

119895=1 119892119895 + 119871 Importantly from (7) in Theorem 1 the signif-icance of the periodic orbit 119909119901(119905) is that within the period119862 and the total number of jobs arriving at the system isexactly equal to the total number of jobs leaving the systemat service rates Specifically if the signalized intersectionis modeled as a switched server system inequality (4) is

the undersaturated condition for signalized intersections andthe period 119862 is the minimum signal cycle (refer to detaileddiscussions in [6]) Moreover the consensus problems (iestates of the system can converge to a common value bylocal protocol) have become fundamental investigations incoordinated control of multiagent systems due to extensiveapplications in engineering fields (eg refer to [18 19])In the sense of traffic control the saturation level of somedirection is defined as the ratio of total number of vehiclesarriving at and leaving the intersection From the significanceof the periodic orbit 119909119901(119905) saturation levels are equal indifferent directions Then the emptying policy can realizethe consensus of saturation levels in traffic control implyingthe balance of traffic loads in different directions Thus theperiodic orbit 119909119901(119905) has practical meanings in applications totraffic control

32 Service-Time-Limited Policy The emptying policy doesnot restrict service-time for buffers However the problemof constraints on service-time of buffers is of importancein practical applications as stated in Introduction In thissubsection the service-time-limited policy is presented basedon emptying policy which can be described by the following

The first two terms (1) and (2) are the same as those indescriptions of emptying policy and (3) in emptying policyis replaced by the following

(31015840) When the buffer 119894 isin 1 119899 is being served theservice-time 119892

119894(119896) for the queue is given by

119892119894 (119896) =

119892119894 If

119909119894(119905119894

119896)

119904119894minus 119902119894

lt 119892119894

119909119894(119905119894

119896)

119904119894minus 119902119894

If 119892119894le

119909119894(119905119894

119896)

119904119894minus 119902119894

le 119892119894

119892119894 If

119909119894(119905119894

119896)

119904119894minus 119902119894

gt 119892119894

(8)

where 119892119894≜ 119892119894minus 119902119894119904minus1119894Γ119894min and 119892

119894≜ 119892119894+ 119902119894119904minus1119894Γ119894max

are respectively the shortest and longest service-timeassigned to the buffer 119894 where 119892

119894is given by (7) and

both Γ119894min and Γ

119894max are respectively called service-time lower-limit and upper-limit factors satisfying0 lt Γ119894min lt 119862 and Γ

119894max gt 0

The information resulted from 119862 and 119892119894of the periodic

orbit 119909119901(119905) determined inTheorem 1 is utilized for the designof service-time-limited policy From (8) the service-time119892119894(119896) of the buffer 119894 within the 119896th cycle is respectively

restricted by the shortest service-time 119892119894and longest service-

time 119892119894 If the queue-emptying time 119909

119894(119905119894

119896)(119904119894minus 119902119894)minus1 is less

than 119892119894assigned to the buffer 119894 then 119892

119894(119896) = 119892

119894 In this case

the serving process of the buffer 119894 is as follows the queuein the buffer 119894 is first served at the service-rate 119904

119894until the

queue is emptied (refer to dynamics in (1)) and then the buffer119894 is served at the arriving-rate 119902

119894until the shortest service-

time 119892119894ends (refer to dynamics in (2)) If the queue-emptying

time 119909119894(119905119894

119896)(119904119894minus 119902119894)minus1 is more than 119892

119894assigned to the buffer 119894

4 Mathematical Problems in Engineering

then 119892119894(119896) = 119892

119894 Otherwise the queue 119909

119894(119905119894

119896) in the buffer 119894 is

emptied and the server switches to the next bufferConsider the following inequality

119899

sum

119895=1119902119895119904minus1119895

lt minmin119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(9)

If Γ119894min and Γ

119894max satisfy 0 lt Γ1min = sdot sdot sdot = Γ119899min lt 119862 and

Γ1max = sdot sdot sdot = Γ119899max gt 0 then (9) is the same as (4)The following results hold for switched server systems

under service-time-limited policy

Theorem 3 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy Assume thatthe total load of buffers satisfies (9) Then any solution to thesystem asymptotically converges to the periodic orbit 119909119901(119905)

The proof of Theorem 3 can be referred to the appendixFurthermore consider the following two special cases forservice-time-limited policy

(C1) Γ1min = sdot sdot sdot = Γ119899min = 0 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 119892119894 119892119894gt 119892119894 119894 = 1 119899

(C2) Γ1min = sdot sdot sdot = Γ119899min = 119862 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 0 119892

119894gt 119892119894 119894 = 1 119899

Consider the following inequality

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(10)

If Γ1max = sdot sdot sdot = Γ119899max gt 0 is satisfied then (10) is the same as(4)

Theorem 4 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with factorssatisfying (C1) or (C2) Assume that the total load of bufferssatisfies (10) Then any solution to the system asymptoticallyconverges to the periodic orbit 119909119901(119905)

Proof When applying service-time-limited policy with fac-tors satisfying (C1) the statements in Theorem 4 can bederived by setting Γ

119894min = 0 119894 = 1 119899 in the proof ofTheorem 3 and when applying service-time-limited policywith factors satisfying (C2) the statements inTheorem 4 canbe derived by Cases 1 and 3 in the proof of Theorem 3 Inabove two cases (9) in the proof of Theorem 3 is changed to(10)

4 Feasibility of Service-Time-Limited Policy

Based on emptying policy service-time-limited policy admitsservice-time constraints on buffers by introducing service-time lower-limit and upper-limit factors Γ

119894min and Γ119894max 119894 =1 119899 but does not bring constraints on the buffer capacity

However the buffer capacity is finite for most of practicalproblems Thus we furthermore consider the execution ofservice-time-limited policy in case of finite buffer capacity

Let 119909max119894

gt 0 119894 = 1 119899 be the capacity of the buffer 119894defined as the maximum queue of jobs that the buffer 119894 canaccommodate Then the admissible region 119872 sub 119883 of thesystem is denoted as119872 ≜ [0 119909max

1 ] times sdot sdot sdot times [0 119909max119899

]It is derived from the significance of the periodic orbit

119909119901(119905) that the maximum queue of jobs in the buffer 119894 is given

by 119902119894(119862 minus 119892

119894) within the period 119862 Assume that the periodic

orbit 119909119901(119905) lies inside the admissible region119872 that is

119909max119894

gt 119902119894(119862minus119892

119894) 119894 = 1 119899 (11)

Definition 5 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policyThe state1199090 isin 119872 is called feasible if for the given service-time lower-limit factors Γ119891

119894min 119894 = 1 119899 there exist service-time upper-limit factors Γ119888

119894max gt 0 119894 = 1 119899 such that the solution119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in 1199090 asymptoticallyconverges to the periodic orbit 119909119901(119905) and moreover satisfies119909(119905) isin 119872 forall119905 ge 0

Furthermore it is deduced from (11) that there must existservice-time upper-limit factors Γ

119894max gt 0 119894 = 1 119899 satis-fying the following inequalities

119909max119894

ge 119902119894[

[

(119862minus119892119894) +sum

119895 =119894

119902119895119904minus1119895Γ119895max]

]

= 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119894 = 1 119899

(12)

Inequalities (12) indicate that when the queue of jobs in thebuffer 119894 is emptied the queue of jobs in the buffer 119894 does notexceed the buffer capacity after one cycle of the server Thefactors Γ

119894max 119894 = 1 119899 satisfying (12) are noted as Γ119888119894max 119894 =

1 119899 in the following parts

Theorem 6 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min and Γ119888

119894max gt 0 119894 = 1 119899 Assume that the total load ofbuffers satisfies (9) (or (10) if all Γ119891

119894min 119894 = 1 119899 satisfy (C1)or (C2)) and the state 1199090 isin 119872 has the property that thesolution 119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in the state 1199090satisfies the condition [1199091(11990511) 119909119899(119905

119899

1)]119879isin 119872 where 1199051198941 is

the moment the server starts serving the buffer 119894 within the firstcycle of the serverThen the state1199090 isin 119872 is feasible with respectto Γ119888119894max gt 0 119894 = 1 119899

Theproof ofTheorem 6 can be referred to the appendix Itis derived from Theorem 6 that the checking conditionfor feasibility of the state is that the corresponding solutiondoes not exceed the buffer capacity within the first cycleof the server with given Γ

119891

119894min and Γ119888

119894max 119894 = 1 119899Accordingly the feasible region 1198830[Γ

119891

119894min Γ119888

119894max] sube 119872 thatis all of feasible states with respect to Γ

119891

119894min and Γ119888

119894max 119894 =

Mathematical Problems in Engineering 5

1 119899 can be obtained from the checking condition forfeasibility of the state Specifically analytic expression offeasible region 1198830[Γ

119891

119894min Γ119888

119894max] for switched server systemswith two buffers can be easily determined as follows

(1) If 0 le Γ119891

119894min lt 119862 119894 = 1 2 then 1198830[Γ119891

119894min Γ119888

119894max] =

11988310 cup 119883

20 cup 119883

30 where 119883

119894

0 119894 = 1 2 3 are respectivelygiven by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1lt 1198921 minus 1199021119904

minus11 Γ119891

1min

1199092 (0) + 1199022 [(1198921 minus 1199021119904minus11 Γ119891

1min) + 11989712] le 119909max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198921 minus 1199021119904minus11 Γ119891

1min le 1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988330 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(13)

(2) If Γ119891119894min = 119862 119894 = 1 2 then1198830[Γ

119891

119894min Γ119888

119894max] = 11988310 cup119883

20

where1198831198940 119894 = 1 2 are respectively given by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(14)

From Theorem 6 feasibility of the state depends onchoices of factors Γ119891

119894min and Γ119888

119894max 119894 = 1 119899 Howeverinfeasibility of the state with respect to some given factorsΓ119891

119894min and Γ119888119894max 119894 = 1 119899 does not imply inexistence of

factors ensuring the state is feasible Furthermore we con-sider the problem of how to solve factors Γ119888

119894max 119894 = 1 119899such that the given state is feasible with given Γ

119891

119894min 119894 =

1 119899If service-time-limited policy is applied with given

Γ119891

119894min = 0 or 0 lt Γ119891119894min lt 119862 119894 = 1 119899 in terms of the check-

ing condition for feasibility of states in Theorem 6 the givenstate 1199090 = [1199091(0) 119909119899(0)]

119879isin 119872 is infeasible if at least one

of the following inequalities holds

119909119894 (0) gt 119909max

119894

minus 119902119894[

[

119894minus1sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) +

119894minus1sum

119895=1119897119895119895+1]

]

119894 = 2 119899

(15)

The significance of (15) is that the queue in the buffer 119894 ge 2exceeds the buffer capacity even if all of buffers 119895 119895 =

1 119894minus1 are servedwith the shortest service-timewithin thefirst cycle of the server Furthermore a sufficient condition isgiven for determining Γ119888

119894max 119894 = 1 119899 ensuring the givenstate 1199090 isin 119872 is feasible

Proposition 7 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min 119894 = 1 119899 For the given state 1199090 isin 119872 if the set Ω(1199090)of factors [Γ1max Γ119899max]

119879 given by (16) is nonempty andthere exists [Γ1198881max Γ

119888

119899max]119879isin Ω(1199090) such that the total

load of buffers satisfies (9) (or (10) if all Γ119891119894min 119894 = 1 119899

satisfy (C1) or (C2)) then the given state 1199090 isin 119872 is feasiblewith respect to [Γ1198881max Γ

119888

119899max]119879

(a)

sum

119895 =1119902119895119904minus1119895Γ119895max le

119909max11199021

minus sum

119895 =1119892119895minus 119871

sum

119895 =119899

119902119895119904minus1119895Γ119895max le

119909max119899

119902119899

minus sum

119895 =119899

119892119895minus 119871

(b)

1199021119904minus11 Γ1max le

[119909max2 minus 1199092 (0)]

1199022minus 1198921 minus 11989712

119899minus1sum

119895=1119902119895119904minus1119895Γ119895max le

[119909max119899

minus 119909119899 (0)]

119902119899

minus

119899minus1sum

119895=1119892119895minus

119899minus1sum

119895=1119897119895119895+1

(c)

0 lt Γ1max le Γlowast

1max

0 lt Γ119899max le Γ

lowast

119899max

(16)

Proof Inequalities (a) in (16) imply that (12) holds and wecan derive from (b) in (16) that

1199092 (11990521) le 1199092 (0) + 1199022 [(1198921 + 1199021119904

minus11 Γ1max) + 11989712]

le 119909max2

119909119899(119905119899

1) le 119909119899 (0)

+ 119902119899[

[

119899minus1sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) +

119899minus1sum

119895=1119897119895119895+1]

]

le 119909max119899

(17)

6 Mathematical Problems in Engineering

which indicates that the solution does not exceed the buffercapacity within the first cycle of the server FurthermoreΓlowast

119894max 119894 = 1 119899 in (c) are the maximum allowable service-time upper-limit factors Thus from Theorem 6 the givenstate 1199090 isin 119872 is feasible

5 Conclusions

For most of real-world problems about queuing systemsservice-times and queues of buffers must be constrained Inthis paper inspired by practical problems in traffic controlthe service-time-limited policy is proposed which is theextension to emptying policy Moreover the execution ofservice-time-limited policy in the case of finite buffer capaci-ties is considered and the notion of feasibility of states underservice-time-limited policy is presented Furthermore basedon the checking condition for feasibility of states (ie thesolution does not exceed buffer capacities within the firstcycle of the server) a sufficient condition for determiningfeasibility of states is given

The scheduling policy proposed in this paper admitstaking into consideration service-time and queue constraintson buffers by the introduction of the notion of feasibility ofstates and service-time upper-limit factors for the feasiblestate can be solved by testing the nonempty set Ω(1199090) Thusour results can be applied to traffic control as stated inthe Introduction especially in critical saturation case forexample the length of queues of vehicles on incoming linksmay be larger with lower traffic loads satisfying (4) Signalcontrol of T-shape intersection is typical application of ourresults which can be referred to [6] for details

From views of traffic control the server may servemultiple nonconflicting flows which is our further researchextension of results in the paper

Appendix

Proof of Theorem 3 Assume that 119905119894119896and 119879

119894

119896 respectively

represent moments that the server starts and finishes servingthe queue in the buffer 119894 in terms of service-time-limitedpolicy within the 119896th cycle of the server forall119894 isin 1 119899 119896 =1 2 Then 1199051198941 is the moment that the server starts servingthe buffer 119894 within the first cycle of the server Considerthe following three possible cases for any solution 119909(119905) =

[1199091(119905) 119909119899(119905)]119879 to the system originating in the initial state

1199090 isin 119883

Case 1 119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894(119905

119894

1)(119904119894 minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max

Case 2 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min

Case3 119909119894(119905119894

1)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1

119894Γ119894max

We prove that the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879

asymptotically converges to the periodic orbit 119909119901(119905) in anycase above

Case 1 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1) (119904119894 minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A1)

then the queue-emptying time 119909119894(119905119894

119898)(119904119894minus 119902119894)minus1 forall119898 ge 1 of

the buffer 119894 within any cycle satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A2)

Proof of Case 1 We prove Case 1 by using mathematicalinduction From (A1) Case 1 holds with119898 = 1 Furthermoreassume that Case 1 holds with some 119898 ge 1 then in terms ofservice-time-limited policy we have that 119909

119894(119879119894

119898) = 0 and

119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

le 119909119894(119905119894

119898+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

(A3)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895minus119902119895119904minus1119895Γ119895min)+119871]

satisfies

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] minus 119902119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

minus

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895min

1 minus 119902119894119904minus1119894

ge 119892119894minus

119902119894119904minus1119894max119895isin1119899 Γ119895minsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A4)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

(A5)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le [1minus 119902119894119904minus1119894]

Γ119894min

max119895isin1119899 Γ119895min

(A6)

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

then 119892119894(119896) = 119892

119894 Otherwise the queue 119909

119894(119905119894

119896) in the buffer 119894 is

emptied and the server switches to the next bufferConsider the following inequality

119899

sum

119895=1119902119895119904minus1119895

lt minmin119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(9)

If Γ119894min and Γ

119894max satisfy 0 lt Γ1min = sdot sdot sdot = Γ119899min lt 119862 and

Γ1max = sdot sdot sdot = Γ119899max gt 0 then (9) is the same as (4)The following results hold for switched server systems

under service-time-limited policy

Theorem 3 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy Assume thatthe total load of buffers satisfies (9) Then any solution to thesystem asymptotically converges to the periodic orbit 119909119901(119905)

The proof of Theorem 3 can be referred to the appendixFurthermore consider the following two special cases forservice-time-limited policy

(C1) Γ1min = sdot sdot sdot = Γ119899min = 0 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 119892119894 119892119894gt 119892119894 119894 = 1 119899

(C2) Γ1min = sdot sdot sdot = Γ119899min = 119862 Γ

119894max gt 0 119894 = 1 119899 thatis 119892119894= 0 119892

119894gt 119892119894 119894 = 1 119899

Consider the following inequality

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(10)

If Γ1max = sdot sdot sdot = Γ119899max gt 0 is satisfied then (10) is the same as(4)

Theorem 4 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with factorssatisfying (C1) or (C2) Assume that the total load of bufferssatisfies (10) Then any solution to the system asymptoticallyconverges to the periodic orbit 119909119901(119905)

Proof When applying service-time-limited policy with fac-tors satisfying (C1) the statements in Theorem 4 can bederived by setting Γ

119894min = 0 119894 = 1 119899 in the proof ofTheorem 3 and when applying service-time-limited policywith factors satisfying (C2) the statements inTheorem 4 canbe derived by Cases 1 and 3 in the proof of Theorem 3 Inabove two cases (9) in the proof of Theorem 3 is changed to(10)

4 Feasibility of Service-Time-Limited Policy

Based on emptying policy service-time-limited policy admitsservice-time constraints on buffers by introducing service-time lower-limit and upper-limit factors Γ

119894min and Γ119894max 119894 =1 119899 but does not bring constraints on the buffer capacity

However the buffer capacity is finite for most of practicalproblems Thus we furthermore consider the execution ofservice-time-limited policy in case of finite buffer capacity

Let 119909max119894

gt 0 119894 = 1 119899 be the capacity of the buffer 119894defined as the maximum queue of jobs that the buffer 119894 canaccommodate Then the admissible region 119872 sub 119883 of thesystem is denoted as119872 ≜ [0 119909max

1 ] times sdot sdot sdot times [0 119909max119899

]It is derived from the significance of the periodic orbit

119909119901(119905) that the maximum queue of jobs in the buffer 119894 is given

by 119902119894(119862 minus 119892

119894) within the period 119862 Assume that the periodic

orbit 119909119901(119905) lies inside the admissible region119872 that is

119909max119894

gt 119902119894(119862minus119892

119894) 119894 = 1 119899 (11)

Definition 5 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policyThe state1199090 isin 119872 is called feasible if for the given service-time lower-limit factors Γ119891

119894min 119894 = 1 119899 there exist service-time upper-limit factors Γ119888

119894max gt 0 119894 = 1 119899 such that the solution119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in 1199090 asymptoticallyconverges to the periodic orbit 119909119901(119905) and moreover satisfies119909(119905) isin 119872 forall119905 ge 0

Furthermore it is deduced from (11) that there must existservice-time upper-limit factors Γ

119894max gt 0 119894 = 1 119899 satis-fying the following inequalities

119909max119894

ge 119902119894[

[

(119862minus119892119894) +sum

119895 =119894

119902119895119904minus1119895Γ119895max]

]

= 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119894 = 1 119899

(12)

Inequalities (12) indicate that when the queue of jobs in thebuffer 119894 is emptied the queue of jobs in the buffer 119894 does notexceed the buffer capacity after one cycle of the server Thefactors Γ

119894max 119894 = 1 119899 satisfying (12) are noted as Γ119888119894max 119894 =

1 119899 in the following parts

Theorem 6 Consider the switched server system described by(1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min and Γ119888

119894max gt 0 119894 = 1 119899 Assume that the total load ofbuffers satisfies (9) (or (10) if all Γ119891

119894min 119894 = 1 119899 satisfy (C1)or (C2)) and the state 1199090 isin 119872 has the property that thesolution 119909(119905) = [1199091(119905) 119909119899(119905)]

119879 originating in the state 1199090satisfies the condition [1199091(11990511) 119909119899(119905

119899

1)]119879isin 119872 where 1199051198941 is

the moment the server starts serving the buffer 119894 within the firstcycle of the serverThen the state1199090 isin 119872 is feasible with respectto Γ119888119894max gt 0 119894 = 1 119899

Theproof ofTheorem 6 can be referred to the appendix Itis derived from Theorem 6 that the checking conditionfor feasibility of the state is that the corresponding solutiondoes not exceed the buffer capacity within the first cycleof the server with given Γ

119891

119894min and Γ119888

119894max 119894 = 1 119899Accordingly the feasible region 1198830[Γ

119891

119894min Γ119888

119894max] sube 119872 thatis all of feasible states with respect to Γ

119891

119894min and Γ119888

119894max 119894 =

Mathematical Problems in Engineering 5

1 119899 can be obtained from the checking condition forfeasibility of the state Specifically analytic expression offeasible region 1198830[Γ

119891

119894min Γ119888

119894max] for switched server systemswith two buffers can be easily determined as follows

(1) If 0 le Γ119891

119894min lt 119862 119894 = 1 2 then 1198830[Γ119891

119894min Γ119888

119894max] =

11988310 cup 119883

20 cup 119883

30 where 119883

119894

0 119894 = 1 2 3 are respectivelygiven by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1lt 1198921 minus 1199021119904

minus11 Γ119891

1min

1199092 (0) + 1199022 [(1198921 minus 1199021119904minus11 Γ119891

1min) + 11989712] le 119909max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198921 minus 1199021119904minus11 Γ119891

1min le 1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988330 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(13)

(2) If Γ119891119894min = 119862 119894 = 1 2 then1198830[Γ

119891

119894min Γ119888

119894max] = 11988310 cup119883

20

where1198831198940 119894 = 1 2 are respectively given by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(14)

From Theorem 6 feasibility of the state depends onchoices of factors Γ119891

119894min and Γ119888

119894max 119894 = 1 119899 Howeverinfeasibility of the state with respect to some given factorsΓ119891

119894min and Γ119888119894max 119894 = 1 119899 does not imply inexistence of

factors ensuring the state is feasible Furthermore we con-sider the problem of how to solve factors Γ119888

119894max 119894 = 1 119899such that the given state is feasible with given Γ

119891

119894min 119894 =

1 119899If service-time-limited policy is applied with given

Γ119891

119894min = 0 or 0 lt Γ119891119894min lt 119862 119894 = 1 119899 in terms of the check-

ing condition for feasibility of states in Theorem 6 the givenstate 1199090 = [1199091(0) 119909119899(0)]

119879isin 119872 is infeasible if at least one

of the following inequalities holds

119909119894 (0) gt 119909max

119894

minus 119902119894[

[

119894minus1sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) +

119894minus1sum

119895=1119897119895119895+1]

]

119894 = 2 119899

(15)

The significance of (15) is that the queue in the buffer 119894 ge 2exceeds the buffer capacity even if all of buffers 119895 119895 =

1 119894minus1 are servedwith the shortest service-timewithin thefirst cycle of the server Furthermore a sufficient condition isgiven for determining Γ119888

119894max 119894 = 1 119899 ensuring the givenstate 1199090 isin 119872 is feasible

Proposition 7 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min 119894 = 1 119899 For the given state 1199090 isin 119872 if the set Ω(1199090)of factors [Γ1max Γ119899max]

119879 given by (16) is nonempty andthere exists [Γ1198881max Γ

119888

119899max]119879isin Ω(1199090) such that the total

load of buffers satisfies (9) (or (10) if all Γ119891119894min 119894 = 1 119899

satisfy (C1) or (C2)) then the given state 1199090 isin 119872 is feasiblewith respect to [Γ1198881max Γ

119888

119899max]119879

(a)

sum

119895 =1119902119895119904minus1119895Γ119895max le

119909max11199021

minus sum

119895 =1119892119895minus 119871

sum

119895 =119899

119902119895119904minus1119895Γ119895max le

119909max119899

119902119899

minus sum

119895 =119899

119892119895minus 119871

(b)

1199021119904minus11 Γ1max le

[119909max2 minus 1199092 (0)]

1199022minus 1198921 minus 11989712

119899minus1sum

119895=1119902119895119904minus1119895Γ119895max le

[119909max119899

minus 119909119899 (0)]

119902119899

minus

119899minus1sum

119895=1119892119895minus

119899minus1sum

119895=1119897119895119895+1

(c)

0 lt Γ1max le Γlowast

1max

0 lt Γ119899max le Γ

lowast

119899max

(16)

Proof Inequalities (a) in (16) imply that (12) holds and wecan derive from (b) in (16) that

1199092 (11990521) le 1199092 (0) + 1199022 [(1198921 + 1199021119904

minus11 Γ1max) + 11989712]

le 119909max2

119909119899(119905119899

1) le 119909119899 (0)

+ 119902119899[

[

119899minus1sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) +

119899minus1sum

119895=1119897119895119895+1]

]

le 119909max119899

(17)

6 Mathematical Problems in Engineering

which indicates that the solution does not exceed the buffercapacity within the first cycle of the server FurthermoreΓlowast

119894max 119894 = 1 119899 in (c) are the maximum allowable service-time upper-limit factors Thus from Theorem 6 the givenstate 1199090 isin 119872 is feasible

5 Conclusions

For most of real-world problems about queuing systemsservice-times and queues of buffers must be constrained Inthis paper inspired by practical problems in traffic controlthe service-time-limited policy is proposed which is theextension to emptying policy Moreover the execution ofservice-time-limited policy in the case of finite buffer capaci-ties is considered and the notion of feasibility of states underservice-time-limited policy is presented Furthermore basedon the checking condition for feasibility of states (ie thesolution does not exceed buffer capacities within the firstcycle of the server) a sufficient condition for determiningfeasibility of states is given

The scheduling policy proposed in this paper admitstaking into consideration service-time and queue constraintson buffers by the introduction of the notion of feasibility ofstates and service-time upper-limit factors for the feasiblestate can be solved by testing the nonempty set Ω(1199090) Thusour results can be applied to traffic control as stated inthe Introduction especially in critical saturation case forexample the length of queues of vehicles on incoming linksmay be larger with lower traffic loads satisfying (4) Signalcontrol of T-shape intersection is typical application of ourresults which can be referred to [6] for details

From views of traffic control the server may servemultiple nonconflicting flows which is our further researchextension of results in the paper

Appendix

Proof of Theorem 3 Assume that 119905119894119896and 119879

119894

119896 respectively

represent moments that the server starts and finishes servingthe queue in the buffer 119894 in terms of service-time-limitedpolicy within the 119896th cycle of the server forall119894 isin 1 119899 119896 =1 2 Then 1199051198941 is the moment that the server starts servingthe buffer 119894 within the first cycle of the server Considerthe following three possible cases for any solution 119909(119905) =

[1199091(119905) 119909119899(119905)]119879 to the system originating in the initial state

1199090 isin 119883

Case 1 119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894(119905

119894

1)(119904119894 minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max

Case 2 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min

Case3 119909119894(119905119894

1)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1

119894Γ119894max

We prove that the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879

asymptotically converges to the periodic orbit 119909119901(119905) in anycase above

Case 1 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1) (119904119894 minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A1)

then the queue-emptying time 119909119894(119905119894

119898)(119904119894minus 119902119894)minus1 forall119898 ge 1 of

the buffer 119894 within any cycle satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A2)

Proof of Case 1 We prove Case 1 by using mathematicalinduction From (A1) Case 1 holds with119898 = 1 Furthermoreassume that Case 1 holds with some 119898 ge 1 then in terms ofservice-time-limited policy we have that 119909

119894(119879119894

119898) = 0 and

119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

le 119909119894(119905119894

119898+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

(A3)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895minus119902119895119904minus1119895Γ119895min)+119871]

satisfies

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] minus 119902119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

minus

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895min

1 minus 119902119894119904minus1119894

ge 119892119894minus

119902119894119904minus1119894max119895isin1119899 Γ119895minsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A4)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

(A5)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le [1minus 119902119894119904minus1119894]

Γ119894min

max119895isin1119899 Γ119895min

(A6)

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

1 119899 can be obtained from the checking condition forfeasibility of the state Specifically analytic expression offeasible region 1198830[Γ

119891

119894min Γ119888

119894max] for switched server systemswith two buffers can be easily determined as follows

(1) If 0 le Γ119891

119894min lt 119862 119894 = 1 2 then 1198830[Γ119891

119894min Γ119888

119894max] =

11988310 cup 119883

20 cup 119883

30 where 119883

119894

0 119894 = 1 2 3 are respectivelygiven by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1lt 1198921 minus 1199021119904

minus11 Γ119891

1min

1199092 (0) + 1199022 [(1198921 minus 1199021119904minus11 Γ119891

1min) + 11989712] le 119909max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198921 minus 1199021119904minus11 Γ119891

1min le 1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988330 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(13)

(2) If Γ119891119894min = 119862 119894 = 1 2 then1198830[Γ

119891

119894min Γ119888

119894max] = 11988310 cup119883

20

where1198831198940 119894 = 1 2 are respectively given by

11988310 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1le 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [1199091 (0) (1199041 minus 1199021)minus1+ 11989712] le 119909

max2

11988320 =

1199090 isin119872

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091 (0) (1199041 minus 1199021)minus1gt 1198921 + 1199021119904

minus11 Γ119888

1max

1199092 (0) + 1199022 [(1198921 + 1199021119904minus11 Γ119888

1max) + 11989712] le 119909max2

(14)

From Theorem 6 feasibility of the state depends onchoices of factors Γ119891

119894min and Γ119888

119894max 119894 = 1 119899 Howeverinfeasibility of the state with respect to some given factorsΓ119891

119894min and Γ119888119894max 119894 = 1 119899 does not imply inexistence of

factors ensuring the state is feasible Furthermore we con-sider the problem of how to solve factors Γ119888

119894max 119894 = 1 119899such that the given state is feasible with given Γ

119891

119894min 119894 =

1 119899If service-time-limited policy is applied with given

Γ119891

119894min = 0 or 0 lt Γ119891119894min lt 119862 119894 = 1 119899 in terms of the check-

ing condition for feasibility of states in Theorem 6 the givenstate 1199090 = [1199091(0) 119909119899(0)]

119879isin 119872 is infeasible if at least one

of the following inequalities holds

119909119894 (0) gt 119909max

119894

minus 119902119894[

[

119894minus1sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) +

119894minus1sum

119895=1119897119895119895+1]

]

119894 = 2 119899

(15)

The significance of (15) is that the queue in the buffer 119894 ge 2exceeds the buffer capacity even if all of buffers 119895 119895 =

1 119894minus1 are servedwith the shortest service-timewithin thefirst cycle of the server Furthermore a sufficient condition isgiven for determining Γ119888

119894max 119894 = 1 119899 ensuring the givenstate 1199090 isin 119872 is feasible

Proposition 7 Consider the switched server system describedby (1) (2) and (3) under service-time-limited policy with givenΓ119891

119894min 119894 = 1 119899 For the given state 1199090 isin 119872 if the set Ω(1199090)of factors [Γ1max Γ119899max]

119879 given by (16) is nonempty andthere exists [Γ1198881max Γ

119888

119899max]119879isin Ω(1199090) such that the total

load of buffers satisfies (9) (or (10) if all Γ119891119894min 119894 = 1 119899

satisfy (C1) or (C2)) then the given state 1199090 isin 119872 is feasiblewith respect to [Γ1198881max Γ

119888

119899max]119879

(a)

sum

119895 =1119902119895119904minus1119895Γ119895max le

119909max11199021

minus sum

119895 =1119892119895minus 119871

sum

119895 =119899

119902119895119904minus1119895Γ119895max le

119909max119899

119902119899

minus sum

119895 =119899

119892119895minus 119871

(b)

1199021119904minus11 Γ1max le

[119909max2 minus 1199092 (0)]

1199022minus 1198921 minus 11989712

119899minus1sum

119895=1119902119895119904minus1119895Γ119895max le

[119909max119899

minus 119909119899 (0)]

119902119899

minus

119899minus1sum

119895=1119892119895minus

119899minus1sum

119895=1119897119895119895+1

(c)

0 lt Γ1max le Γlowast

1max

0 lt Γ119899max le Γ

lowast

119899max

(16)

Proof Inequalities (a) in (16) imply that (12) holds and wecan derive from (b) in (16) that

1199092 (11990521) le 1199092 (0) + 1199022 [(1198921 + 1199021119904

minus11 Γ1max) + 11989712]

le 119909max2

119909119899(119905119899

1) le 119909119899 (0)

+ 119902119899[

[

119899minus1sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) +

119899minus1sum

119895=1119897119895119895+1]

]

le 119909max119899

(17)

6 Mathematical Problems in Engineering

which indicates that the solution does not exceed the buffercapacity within the first cycle of the server FurthermoreΓlowast

119894max 119894 = 1 119899 in (c) are the maximum allowable service-time upper-limit factors Thus from Theorem 6 the givenstate 1199090 isin 119872 is feasible

5 Conclusions

For most of real-world problems about queuing systemsservice-times and queues of buffers must be constrained Inthis paper inspired by practical problems in traffic controlthe service-time-limited policy is proposed which is theextension to emptying policy Moreover the execution ofservice-time-limited policy in the case of finite buffer capaci-ties is considered and the notion of feasibility of states underservice-time-limited policy is presented Furthermore basedon the checking condition for feasibility of states (ie thesolution does not exceed buffer capacities within the firstcycle of the server) a sufficient condition for determiningfeasibility of states is given

The scheduling policy proposed in this paper admitstaking into consideration service-time and queue constraintson buffers by the introduction of the notion of feasibility ofstates and service-time upper-limit factors for the feasiblestate can be solved by testing the nonempty set Ω(1199090) Thusour results can be applied to traffic control as stated inthe Introduction especially in critical saturation case forexample the length of queues of vehicles on incoming linksmay be larger with lower traffic loads satisfying (4) Signalcontrol of T-shape intersection is typical application of ourresults which can be referred to [6] for details

From views of traffic control the server may servemultiple nonconflicting flows which is our further researchextension of results in the paper

Appendix

Proof of Theorem 3 Assume that 119905119894119896and 119879

119894

119896 respectively

represent moments that the server starts and finishes servingthe queue in the buffer 119894 in terms of service-time-limitedpolicy within the 119896th cycle of the server forall119894 isin 1 119899 119896 =1 2 Then 1199051198941 is the moment that the server starts servingthe buffer 119894 within the first cycle of the server Considerthe following three possible cases for any solution 119909(119905) =

[1199091(119905) 119909119899(119905)]119879 to the system originating in the initial state

1199090 isin 119883

Case 1 119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894(119905

119894

1)(119904119894 minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max

Case 2 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min

Case3 119909119894(119905119894

1)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1

119894Γ119894max

We prove that the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879

asymptotically converges to the periodic orbit 119909119901(119905) in anycase above

Case 1 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1) (119904119894 minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A1)

then the queue-emptying time 119909119894(119905119894

119898)(119904119894minus 119902119894)minus1 forall119898 ge 1 of

the buffer 119894 within any cycle satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A2)

Proof of Case 1 We prove Case 1 by using mathematicalinduction From (A1) Case 1 holds with119898 = 1 Furthermoreassume that Case 1 holds with some 119898 ge 1 then in terms ofservice-time-limited policy we have that 119909

119894(119879119894

119898) = 0 and

119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

le 119909119894(119905119894

119898+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

(A3)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895minus119902119895119904minus1119895Γ119895min)+119871]

satisfies

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] minus 119902119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

minus

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895min

1 minus 119902119894119904minus1119894

ge 119892119894minus

119902119894119904minus1119894max119895isin1119899 Γ119895minsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A4)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

(A5)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le [1minus 119902119894119904minus1119894]

Γ119894min

max119895isin1119899 Γ119895min

(A6)

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

which indicates that the solution does not exceed the buffercapacity within the first cycle of the server FurthermoreΓlowast

119894max 119894 = 1 119899 in (c) are the maximum allowable service-time upper-limit factors Thus from Theorem 6 the givenstate 1199090 isin 119872 is feasible

5 Conclusions

For most of real-world problems about queuing systemsservice-times and queues of buffers must be constrained Inthis paper inspired by practical problems in traffic controlthe service-time-limited policy is proposed which is theextension to emptying policy Moreover the execution ofservice-time-limited policy in the case of finite buffer capaci-ties is considered and the notion of feasibility of states underservice-time-limited policy is presented Furthermore basedon the checking condition for feasibility of states (ie thesolution does not exceed buffer capacities within the firstcycle of the server) a sufficient condition for determiningfeasibility of states is given

The scheduling policy proposed in this paper admitstaking into consideration service-time and queue constraintson buffers by the introduction of the notion of feasibility ofstates and service-time upper-limit factors for the feasiblestate can be solved by testing the nonempty set Ω(1199090) Thusour results can be applied to traffic control as stated inthe Introduction especially in critical saturation case forexample the length of queues of vehicles on incoming linksmay be larger with lower traffic loads satisfying (4) Signalcontrol of T-shape intersection is typical application of ourresults which can be referred to [6] for details

From views of traffic control the server may servemultiple nonconflicting flows which is our further researchextension of results in the paper

Appendix

Proof of Theorem 3 Assume that 119905119894119896and 119879

119894

119896 respectively

represent moments that the server starts and finishes servingthe queue in the buffer 119894 in terms of service-time-limitedpolicy within the 119896th cycle of the server forall119894 isin 1 119899 119896 =1 2 Then 1199051198941 is the moment that the server starts servingthe buffer 119894 within the first cycle of the server Considerthe following three possible cases for any solution 119909(119905) =

[1199091(119905) 119909119899(119905)]119879 to the system originating in the initial state

1199090 isin 119883

Case 1 119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894(119905

119894

1)(119904119894 minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max

Case 2 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min

Case3 119909119894(119905119894

1)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1

119894Γ119894max

We prove that the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879

asymptotically converges to the periodic orbit 119909119901(119905) in anycase above

Case 1 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1) (119904119894 minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A1)

then the queue-emptying time 119909119894(119905119894

119898)(119904119894minus 119902119894)minus1 forall119898 ge 1 of

the buffer 119894 within any cycle satisfies

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A2)

Proof of Case 1 We prove Case 1 by using mathematicalinduction From (A1) Case 1 holds with119898 = 1 Furthermoreassume that Case 1 holds with some 119898 ge 1 then in terms ofservice-time-limited policy we have that 119909

119894(119879119894

119898) = 0 and

119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

le 119909119894(119905119894

119898+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

(A3)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895minus119902119895119904minus1119895Γ119895min)+119871]

satisfies

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] minus 119902119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

minus

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895min

1 minus 119902119894119904minus1119894

ge 119892119894minus

119902119894119904minus1119894max119895isin1119899 Γ119895minsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A4)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

(A5)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

le [1minus 119902119894119904minus1119894]

Γ119894min

max119895isin1119899 Γ119895min

(A6)

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Substitute (A6) into (A4) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

ge

119902119894[sum119895 =119894(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

119904119894minus 119902119894

ge 119892119894minus 119902119894119904minus1119894Γ119894min

(A7)

The emptying time for the queue 119902119894[sum119895 =119894(119892119895+119902119895119904minus1119895Γ119895max)+119871]

satisfies

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

=

119902119894[sum119895 =119894119892119895+ 119871] + 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119902119894[119862 minus 119892

119894] + 119902119894sum119895 =119894119902119895119904minus1119895Γ119895max

119904119894minus 119902119894

=

119904119894119892119894minus 119902119894119892119894

119904119894minus 119902119894

+

119902119894119904minus1119894sum119895 =119894119902119895119904minus1119895Γ119895max

1 minus 119902119894119904minus1119894

le 119892119894+

119902119894119904minus1119894max119895isin1119899 Γ119895maxsum119895 =119894 119902119895119904

minus1119895

1 minus 119902119894119904minus1119894

(A8)

From (9) we have that

sum

119895 =119894

119902119895119904minus1119895+ 119902119894119904minus1119894

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le

119899

sum

119895=1119902119895119904minus1119895lt

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

(A9)

Then

sum

119895 =119894

119902119895119904minus1119895lt [1minus 119902

119894119904minus1119894]

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

le [1minus 119902119894119904minus1119894]

Γ119894max

max119895isin1119899 Γ119895max

(A10)

Substitute (A10) into (A8) we have that

119909119894(119905119894

119898+1)

119904119894minus 119902119894

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A11)

Thus from (A7) and (A11) Case 1 holds with 119898 + 1 ThenCase 1 holds by induction Here the proof of Case 1 ends

Case 2 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A12)

then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A13)

Proof of Case 2 After one cycle of the server from time 1199051198941 wehave that

119909119894(119905119894

2) = 119909119894 (119905119894

1) +119860 119894 (119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2) (A14)

where 119860119894(119905119894

1 119905119894

2) gt 0 and 119863119894(119905119894

1 119905119894

2) gt 0 are total amountsof jobs arriving at and leaving the buffer 119894 within one cyclerespectively

From (A12) 119860119894(119905119894

1 119905119894

2) and 119863119894(119905119894

1 119905119894

2) in (A14) respec-tively satisfy

119860119894(119905119894

1 119905119894

2) ge 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

119863119894(119905119894

1 119905119894

2) lt 119904119894 (119892119894 minus 119902119894119904minus1119894Γ119894min)

(A15)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

gt 119902119894[

[

119899

sum

119895=1(119892119895minus 119902119895119904minus1119895Γ119895min) + 119871]

]

minus 119904119894(119892119894minus 119902119894119904minus1119894Γ119894min)

=[

[

119902119894(

119899

sum

119895=1119892119895+119871)minus 119904

119894119892119894]

]

+119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

= 119902119894[

[

Γ119894min minus

119899

sum

119895=1119902119895119904minus1119895Γ119895min]

]

ge 119902119894max119895isin1119899

Γ119895min

sdot[

[

min119895isin1119899 Γ119895min

max119895isin1119899 Γ119895min

minus

119899

sum

119895=1119902119895119904minus1119895]

]

(A16)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) gt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1gt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min 119901 = 1 119896 (A17)

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic increasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119894min (A18)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A19)

In terms of service-time-limited policy (A18) and (A8) wehave that

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1

le

119902119894[sum119895 =119894(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

119904119894minus 119902119894

le 119892119894+ 119902119894119904minus1119894Γ119894max

(A20)

Then Case 2 can be obtained from (A19) (A20) and resultsin Case 1 Here the end of proof of Case 2

Case 3 If the queue-emptying time 119909119894(119905119894

1)(119904119894 minus 119902119894)minus1 of the

buffer 119894 satisfies

119909119894(119905119894

1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A21)

Then there must exist 1198960 gt 1 such that

119892119894minus 119902119894119904minus1119894Γ119894min le 119909119894 (119905

119894

1198960+119898) (119904119894minus 119902119894)minus1

le 119892119894+ 119902119894119904minus1119894Γ119894max forall119898 ge 0

(A22)

Proof of Case 3 After one cycle of the server from time 1199051198941 interms of (A21) and service-time-limited policy119860

119894(119905119894

1 119905119894

2) and119863119894(119905119894

1 119905119894

2) in (A14) respectively satisfy

119860119894(119905119894

1 119905119894

2) le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

119863119894(119905119894

1 119905119894

2) = 119904119894 (119892119894 + 119902119894119904minus1119894Γ119894max)

(A23)

Then the increment 119860119894(119905119894

1 119905119894

2) minus 119863119894(119905119894

1 119905119894

2) in the buffer 119894satisfies

119860119894(119905119894

1 119905119894

2) minus119863119894 (119905119894

1 119905119894

2)

le 119902119894[

[

119899

sum

119895=1(119892119895+ 119902119895119904minus1119895Γ119895max) + 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = [119902119894119862minus 119904119894119892119894]

+ 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

= 119902119894[

[

119899

sum

119895=1119902119895119904minus1119895Γ119895max minus Γ119894max]

]

le 119902119894max119895isin1119899

Γ119895max

sdot[

[

119899

sum

119895=1119902119895119904minus1119895minus

min119895isin1119899 Γ119895max

max119895isin1119899 Γ119895max

]

]

(A24)

Thus from (9) and (A14) we have that119860119894(119905119894

1 119905119894

2)minus119863119894(119905119894

1 119905119894

2) lt

0 and 119909119894(119905119894

2)(119904119894minus119902119894)minus1lt 119909119894(119905119894

1)(119904119894minus119902119894)minus1 From analogous pro-

cedures above we can derive the following conclusions thatif

119909119894(119905119894

119901) (119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max 119901 = 1 119896 (A25)

then 119909119894(119905119894

119901)(119904119894minus 119902119894)minus1119896+1119901=1 is a strictly monotonic decreasing

sequence which indicates that there must exist 1198960 ge 2 suchthat

119909119894(119905119894

1198960minus1) (119904119894 minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119894max (A26)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119894max (A27)

After one cycle of the server from time 1199051198941198960minus1 we have that

119909119894(119905119894

1198960) = 119909119894(119905119894

1198960minus1) +119860 119894 (119905119894

1198960minus1 119905119894

1198960)

minus119863119894(119905119894

1198960minus1 119905119894

1198960)

(A28)

where from (A26) 119860119894(119905119894

1198960minus1 119905119894

1198960) and 119863

119894(119905119894

1198960minus1 119905119894

1198960) respec-

tively satisfy

119860119894(119905119894

1198960minus1 119905119894

1198960) ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min)

+ (119892119894+ 119902119894119904minus1119894Γ119894max) + 119871]

]

119863119894(119905119894

1198960minus1 119905119894

1198960) = 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max)

(A29)

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Then the increment119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960) in the buffer

119894 satisfies

119860119894(119905119894

1198960minus1 119905119894

1198960) minus119863119894(119905119894

1198960minus1 119905119894

1198960)

ge 119902119894[

[

sum

119895 =119894

(119892119895minus 119902119895119904minus1119895Γ119895min) + (119892119894 + 119902119894119904

minus1119894Γ119894max)

+ 119871]

]

minus 119904119894(119892119894+ 119902119894119904minus1119894Γ119894max) = 119902119894Γ119894max [119902119894119904

minus1119894

minus 1] minus 119902119894sum

119895 =119894

119902119895119904minus1119895Γ119895min

(A30)

Then

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge

119902119894Γ119894max [119902119894119904

minus1119894minus 1] minus 119902

119894sum119895 =119894119902119895119904minus1119895Γ119895min

119904119894minus 119902119894

=

119902119894119904minus1119894Γ119894max

1 minus 119902119894119904minus1119894

[119902119894119904minus1119894minus 1]

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

sum

119895 =119894

119902119895119904minus1119895Γ119895min

ge minus 119902119894119904minus1119894Γ119894max

minus

119902119894119904minus1119894

1 minus 119902119894119904minus1119894

max119895isin1119899

Γ119895minsum

119895 =119894

119902119895119904minus1119895

(A31)

Substitute (A6) into (A31) we have that

119860119894(119905119894

1198960minus1 119905119894

1198960) minus 119863119894(119905119894

1198960minus1 119905119894

1198960)

119904119894minus 119902119894

ge minus 119902119894119904minus1119894Γ119894max minus 119902119894119904

minus1119894Γ119894min

(A32)

From (A28) (A26) and (A32)

119909119894(119905119894

1198960) (119904119894minus 119902119894)minus1ge 119892119894minus 119902119894119904minus1119894Γ119894min (A33)

Then Case 3 can be obtained from (A27) (A33) and resultsin Case 1 Here ends the proof of Case 3

In conclusion for any one of three possible cases theservice-time-limited policy converges to emptying policyThus from results in Theorem 1 the solution 119909(119905) = [1199091(119905)

119909119899(119905)]119879 asymptotically converges to the periodic orbit

119909119901(119905)

Proof of Theorem 6 Consider switched server systems underservice-time-limited policy with 0 lt Γ

119891

119894min lt 119862 Γ119888

119894max gt 0119894 = 1 119899 We first prove the following statement

Statement 1 If the state 1199090 isin 119872 has the property stated inTheorem 6 then the condition [1199091(119905

1119896) 119909

119899(119905119899

119896)]119879isin 119872

forall119896 ge 1 holds

Proof of Statement 1 We prove the results in Statement 1by using mathematical induction In the case of 119896 = 1Statement 1 holds because of the property of the state 1199090 isin 119872Furthermore assume that Statement 1 holds for some 119896 = 11989601198960 ge 1 that is [1199091(119905

11198960) 119909

119899(119905119899

1198960)]119879isin 119872 Consider three

possible cases for any buffer 119894 isin 1 119899

Case 1 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1lt 119892119894minus 119902119894119904minus1119894Γ119891

119894min then in terms of service-time-limited policy we have that

119909119894(119905119894

1198960+1) le 119902119894[

[

sum

119895 =119894

(119892119895+ 119902119895119904minus1119895Γ119888

119895max) + 119871]

]

(A34)

It is derived from (12) and (A34) that 119909119894(119905119894

1198960+1) le 119909max119894

Case 2 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119892119894minus 119902119894119904minus1119894Γ119891

119894min le 119909119894 (119905119894

1198960) (119904119894minus 119902119894)minus1le 119892119894+ 119902119894119904minus1119894Γ119888

119894max (A35)

then in terms of service-time-limited policy (A34) stillholds Thus we have that 119909

119894(119905119894

1198960+1) le 119909max119894

Case 3 If the queue-emptying time 119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1 satisfies

119909119894(119905119894

1198960)(119904119894minus 119902119894)minus1gt 119892119894+ 119902119894119904minus1119894Γ119888

119894max then from proof of Case 3in proof ofTheorem 3we have that119909

119894(119905119894

1198960+1) lt 119909119894(119905119894

1198960) le 119909

max119894

In conclusion we have that [1199091(119905

11198960+1) 119909119899(119905

119899

1198960+1)]119879isin

119872 which indicates that Statement 1 holds for 119896 = 1198960 + 1 Bymathematical induction Statement 1 holds for forall119896 ge 1 Hereends the proof of Statement 1

Statement 1 immediately implies 119909(119905) = [1199091(119905)

119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover fromTheorem 3 the solution

119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically converges to the

periodic orbit 119909119901(119905) Then from Definition 5 the state 1199090 isin119872 is feasible

Furthermore Statement 1 still holds for Γ119891

119894min 119894 =

1 119899 satisfying (C1) or (C2) which implies 119909(119905) =

[1199091(119905) 119909119899(119905)]119879isin 119872 forall119905 ge 0 Moreover from Theorem 4

the solution 119909(119905) = [1199091(119905) 119909119899(119905)]119879 asymptotically con-

verges to the periodic orbit 119909119901(119905) Then from Definition 5the state 1199090 isin 119872 is feasible

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Acknowledgments

The authors would like to thank the referees for theirconstructive comments This work is partially supportedby National Natural Science Foundation of China(51308005 and 61374191) and Scientific Research Projectof Beijing Education Committee (PXM2015 014212 000023PXM2015 014212 000018 PXM2015 014212 000019 andPXM2015 014212 000021)

References

[1] A S Matveev and A V Savkin Qualitative Theory of HybridDynamical Systems Birkhauser Boston Mass USA 2000

[2] W P Heemels B De Schutter J Lunze and M Lazar ldquoStabilityanalysis and controller synthesis for hybrid dynamical systemsrdquoPhilosophical Transactions of the Royal Society of London SeriesA vol 368 no 1930 pp 4937ndash4960 2010

[3] J R Perkins and P R Kumar ldquoStable distributed real-timescheduling of flexible manufacturingassemblydisassemblysystemsrdquo IEEE Transactions on Automatic Control vol 34 no2 pp 139ndash148 1989

[4] J R Perkins C Humes Jr and P R Kumar ldquoDistributedscheduling of flexible manufacturing systems stability andperformancerdquo IEEE Transactions on Robotics and Automationvol 10 no 2 pp 133ndash141 1994

[5] Y-Z Chen H-F Li and J Ni ldquoModeling and analysis of cycliclinear differential automata for T-intersection signal timingrdquoControl Theory amp Applications vol 28 no 12 pp 1773ndash17782011

[6] ZHHe Y Z Chen J J Shi XGHan andXWu ldquoSteady-statecontrol for signalized intersections modeled as switched serversystemrdquo in Proceedings of the American Control Conference(ACC 13) pp 842ndash847 Washington DC USA June 2013

[7] M A A Boon I J B F Adan E M M Winands and DG Down ldquoDelays at signalized intersections with exhaustivetraffic controlrdquo Probability in the Engineering and InformationalSciences vol 26 no 3 pp 337ndash373 2012

[8] MAA Boon RD vanderMei andEMMWinands ldquoAppli-cations of polling systemsrdquo Surveys in Operations Research andManagement Science vol 16 no 2 pp 67ndash82 2011

[9] A V Savkin and A S Matveev ldquoCyclic linear differential auto-mata a simple class of hybrid dynamical systemsrdquo Automaticavol 36 no 5 pp 727ndash734 2000

[10] Z G Li Y C Soh and C Y Wen Switched and Impulsive Sys-tems Analysis Design and Applications Springer Berlin Ger-many 2005

[11] Z-H He Y-Z Chen and J-J Shi ldquoStability of switched serversystem and signal timing of intersectionrdquo Control Theory ampApplications vol 30 no 2 pp 194ndash200 2013

[12] A V Savkin and J Somlo ldquoOptimal distributed real-timescheduling of flexible manufacturing networks modeled ashybrid dynamical systemsrdquo Robotics and Computer-IntegratedManufacturing vol 25 no 3 pp 597ndash609 2009

[13] E Lefeber and J E Rooda ldquoController design for switchedlinear systems with setupsrdquo Physica A Statistical Mechanics andIts Applications vol 363 no 1 pp 48ndash61 2006

[14] E Lefeber and J E Rooda ldquoController design for flow networksof switched servers with setup times the Kumar-Seidman caseas an illustrative examplerdquo Asian Journal of Control vol 10 no1 pp 55ndash66 2008

[15] V Feoktistova AMatveev E Lefeber and J E Rooda ldquoDesignsof optimal switching feedback decentralized control policies forfluid queueing networksrdquo Mathematics of Control Signals andSystems vol 24 no 4 pp 477ndash503 2012

[16] J A W M van Eekelen E Lefeber and J E Rooda ldquoFeedbackcontrol of 2-product server with setups and bounded buffersrdquoin Proceedings of the American Control Conference pp 544ndash5492006

[17] C Diakaki M Papageorgiou and K Aboudolas ldquoA multivar-iable regulator approach to traffic-responsive network-wide sig-nal controlrdquo Control Engineering Practice vol 10 no 2 pp 183ndash195 2002

[18] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[19] Y R Ge Y Z Chen Y X Zhang and Z H He ldquoState consensusanalysis and design for high-order discrete-time linear multia-gent systemsrdquoMathematical Problems in Engineering vol 2013Article ID 192351 13 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of