research article m/m/1 multiple vacation queueing...

Research ArticleM/M/1 Multiple Vacation Queueing Systems withDifferentiated Vacations

Oliver C. Ibe and Olubukola A. Isijola

Department of Electrical and Computer Engineering, University ofMassachusetts, Lowell, 1 University Avenue, Lowell, MA 01854, USA

Correspondence should be addressed to Oliver C. Ibe; oliver [email protected]

Received 12 March 2014; Revised 12 June 2014; Accepted 17 June 2014; Published 23 July 2014

Academic Editor: Tadashi Dohi

Copyright © 2014 O. C. Ibe and O. A. Isijola. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We consider a multiple vacation queueing system in which a vacation following a busy period has a different distribution from avacation that is takenwithout serving at least one customer. For ease of analysis it is assumed that the service times are exponentiallydistributed and the two vacation types are also exponentially distributed but with different means. The steady-state solution isobtained.

1. Introduction

A vacation queueing system is one in which a server maybecome unavailable for a random period of time from aprimary service center. The time away from the primaryservice center is called a vacation, and it can be the result ofmany factors. In some cases the vacation can be the resultof server breakdown, which means that the system mustbe repaired and brought back to service. It can also be adeliberate action taken to utilize the server in a secondaryservice center when there are no customers present at theprimary service center. Thus, server vacations are useful forthose systems in which the server wishes to utilize his idletime for different purposes, and this makes the queueingmodel be applicable to a variety of real world stochasticservice systems.

Queueing systems with server vacations have attractedthe attention of many researchers since the idea was firstdiscussed in the paper of Levy and Yechiali [1]. Severalexcellent surveys on these vacation models have been doneby Doshi [2, 3], and the books by Takagi [4] and Tian andZhang [5] are devoted to the subject.

There are different types of vacation queueing systems.In the single vacation scheme, the server takes a vacation ofa random duration when the queue is empty. At the end ofthe vacation the server returns to the queue. If there is at least

one customer waiting when the server returns from vacation,the server performs one of the following actions dependingon the service policy.

(a) Under the exhaustive service policy, the server willserve all waiting customers as well as those that arrivewhile he is still serving at the station.He takes anothervacation when the queue becomes empty.

(b) Under the gated service policy, the server will serveonly those customers that he finds at the queue uponhis return from vacation. At the end of their servicethe server will commence another vacation and anycustomers that arrive while the server was alreadyserving at the station will be served when the serverreturns from the vacation.

(c) Under the limited service policy, the server will serveonly a predefined maximum number of customersand thenwill commence another vacation.The single-service scheme in which exactly one customer isserved is a special type of this policy.

If the queue is empty on the server’s return, the server waitsto complete a busy period using one of the service policiesbefore taking another vacation.

In the multiple vacation scheme, if the server returnsfrom a vacation and finds the queue empty, he immediately

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014, Article ID 158247, 6 pageshttp://dx.doi.org/10.1155/2014/158247

2 Modelling and Simulation in Engineering

commences another vacation. If there is at least one waitingcustomer, then he will commence service according to theprevailing service policy.

Observe that in the vacation queueing system we havedescribed the server completely stops service or is switchedoff when he is on vacation. Recently, Servi and Finn [6]introduced the working vacation scheme, in which the serverworks at a different rate rather than completely stoppingservice during a vacation. They applied the M/M/1 queuewith multiple working vacations to model a Wavelength-Division Multiplexing optical access network and derivedthe probability-generating function (PGF) of the number ofcustomers in the system. In the original formulation of theworking vacation scheme the server cannot be interruptedwhen he is on vacation; he resumes full service only whenhis vacation ends.

The working vacation scheme has attracted a lot ofresearch effort, and several authors have extended the originalmodel. Wu and Takagi [7] generalized the model in [6] toan M/G/1 queue with general working vacations. Baba [8]studied a GI/M/1 queue with working vacations by usingthe matrix analytic method. Banik et al. [9] analyzed theGI/M/1/N queue with working vacations. Liu et al. [10]established a stochastic decomposition result in the M/M/1queue with working vacations.

For the batch arrival queues, Xu et al. [11] studied a batcharrival M𝑋/M/1 queue with single working vacation. Usingthe matrix analytic method, they derived the PGF of the sta-tionary system length distribution. Baba [12] studied a batcharrival M𝑋/M/1 queue with multiple working vacations. Heobtained the PGF of the stationary system length distributionand the stochastic decomposition structure of system lengththat indicates the relationship with that of M𝑋/M/1 queuewithout vacation.

Some researchers have also considered discrete-timeworking vacation systems. Tian et al. [13] considered the dis-crete timeGeo/Geo/1 queue withmultiple working vacations.Li and Tian [14] analyzed the discrete-time Geo/Geo/1 queuewith single working vacation. Gao and Liu [15] analyzed theperformance of a discrete-time Geo𝑋/G/1 queue with singleworking vacation. Li et al. [16] discussed a discrete-time batcharrival Geo𝑋/GI/1 queue with working vacations.

Li and Tian [17] analyzed a GI/Geo/1 queue with workingvacations and vacation interruption. Under such a policy, theserver can come back to the normal working level before thevacation ends. They obtained the steady-state distributionsfor the number of customers in the system at arrival epochsand waiting time for an arbitrary customer using the matrix-geometric solution method. The authors also extended themodel to the M/M/1 queue with working vacation andvacation interruptions [18]. The GI/M/1 queue with workingvacations and vacation interruption was studied by Li etal. [19]. Similarly, Zhang and Hou [20] discussed an M/G/1queue with multiple working vacations and vacation inter-ruption.

Recall that in the multiple vacation queueing system it isassumed that the vacation times are independent and identi-cally distributed. However, there are practical environments

where this assumption may not be valid. Specifically, avacation taken after “a hard day’s work” during which manycustomers have been served may be longer than a vacationtaken after the server returns from vacation and finds thequeue empty. We define a vacation queueing system thatdistinguishes between two kinds of vacations that a servercan take as a vacation queueing system with differentiatedvacations. The analysis of the M/M/1 version of this type ofvacation queueing system is the subject of this paper.

Thus, the paper deals with an M/M/1 queueing system inwhich two types of vacations can be taken by the server: avacation taken immediately after the server has finished serv-ing at least one customer and a vacation taken immediatelyafter the server has just returned from a previous vacation tofind that there are no customers waiting.

The model is motivated by certain aspects of human andphysical system behavior. For example, a computer systemcan suffer one of two types of failures: permanent failureand intermittent failure [21]. A permanent failure, which issometimes called a hard failure, requires the physical repairof the failed system, which usually takes a long time becauseit requires the presence of the field services personnel. Bycontrast, after a system has suffered an intermittent failure(or soft failure), no physical repair is required. The system isrestored to operation by means of a system reboot or someother repair function that does not require the presence ofthe field services personnel. As long as the system is not beingused for the intended service, it can be modeled as being onvacation.Thus, vacations associatedwith intermittent failuresare generally of shorter durations than vacations associatedwith permanent failures.

Another example is the following. Consider a gas stationattendant who operates under the following policy. Whenthere is no customer waiting to be attended he will takea break that he can use to perform other functions at thestation. At the end of the break if there is still no waitingcustomer, he will take another break, but if there is at leastone waiting customer, he will serve exhaustively and will takea break when all customers have been served. This is thetraditional multiple vacation model. Suppose now that thereare two types of breaks that he can take. Specifically, afterserving all customers in a busy period that includes at leastone customer, he will take a coffee break or a personal breakwhose length has a given distribution. If he returns froma break and there is no waiting customer, he goes back onanother break whose length has another distribution. Thistime can be used to attend to other duties at the station andusually has a shorter mean duration than the coffee/personalbreak. Thus, long breaks are associated with the completionof a busy period with at least one service completion whileshort breaks are associated with busy periods of zero length.

In general, differentiated vacations occur in environ-ments where “breaks” of different durations can occur. In thispaper we have associated these breaks with the durations ofbusy periods.

Note that this model is different from the traditionalmultiple vacation model because in the traditional multiplevacation model the durations of vacations are identicallydistributed and are independent of the number of customers

Modelling and Simulation in Engineering 3

0, 2 1, 2

1, 0 2, 0 3, 0

3, 12, 11, 10, 1

2, 2 3, 2

𝜇

𝜇

𝜇 𝜇

𝜆

𝜆 𝜆 𝜆 𝜆

𝜆 𝜆 𝜆 𝜆

𝜆 𝜆

𝛾2

𝛾1

𝛾1 𝛾1 𝛾1

𝛾2 𝛾2

· · ·

· · ·

· · ·

Figure 1: State-transition-rate diagram.

served in the busy period preceding the vacation. In thedifferentiated vacation model that we are proposing, thereare two distributions of the durations of vacations: one isassociated with a vacation taken after a nonzero busy periodand the other is associated with a vacation taken after a zerobusy period. The practical application of this model is thatdurations of vacations taken after a nonzero busy period canbe longer than those that are taken when the server did notserve any customer prior to the vacation in order to give theserver a sufficient time to rest following some hectic busyperiod.

The paper is organized as follows. The model is moreformally defined in Section 2. Steady-state analysis of themodel is given in Section 3, computational results are givenin Section 4, and concluding remarks are made in Section 5.

2. System Model

We consider a multiple vacation queueing system wherecustomers arrive according to a Poisson process with rate 𝜆.The time to serve a customer is assumed to be exponentiallydistributed with mean 1/𝜇, where 𝜇 > 𝜆. We assume thatthere are two types of vacations: type 1 vacation that is takenafter a busy period of nonzero duration, and type 2 vacationthat is taken when no customers are waiting for the serverwhen it returns from a vacation. For ease of analysis weassume that the durations of type 1 vacations are independentof the busy period and are exponentially distributed withmean 1/𝛾

1. (As discussed earlier, there are cases where

this independence assumption is not valid. However, wemake this assumption to simplify the analysis.) Similarly,durations of type 2 vacations are assumed to be exponentiallydistributed with mean 1/𝛾

2.

Let the state of the system be denoted by (𝑟, 𝑘), where𝑟 is the number of customers in the system, 𝑘 = 0 if theserver is active serving customers, 𝑘 = 1 if the server is ona type 1 vacation, and 𝑘 = 2 if the server is on a type 2vacation. Thus, the system can be modeled by a continuous-time Markov chain whose state-transition-rate diagram isshown in Figure 1.

3. Steady-State Analysis

Let 𝑝𝑛,𝑘

(𝑡) denote the probability that the process is in state(𝑛, 𝑘) at time 𝑡, and let

𝑝𝑛,𝑘

= lim𝑡→∞

𝑝𝑛,𝑘 (𝑡) . (1)

The main result of the paper is stated via the followingtheorem.

Theorem 1. The steady-state probability 𝑝𝑛,𝑘

is given by

𝑝𝑛,𝑘

=

{{{{{{{{{{

{{{{{{{{{{

{

𝜌[𝛼1𝛽1(𝛽𝑛−1

1− 𝜌𝑛−1

)

𝛽1− 𝜌

+𝛼2𝛽2(𝛽𝑛−1

2− 𝜌𝑛−1

)

𝛽2− 𝜌

+ 𝜌𝑛−2

]𝑝1,0

𝑘 = 0

𝛼1𝛽𝑛

1𝑝1,0

𝑘 = 1

𝛼2𝛽𝑛

2𝑝1,0

𝑘 = 2,

(2)

where𝑝1,0

= ((1 − 𝜌) (1 − 𝛽1) (1 − 𝛽

2))

× ((1 − 𝛽1) (1 − 𝛽

2) + 𝛼1(1 − 𝛽

2)

× {1 − 𝜌 (1 − 𝛽1)} + 𝛼

2(1 − 𝛽

1)

× {1 − 𝜌 (1 − 𝛽2)})−1,

(3)

𝜌 = 𝜆/𝜇 is the offered load, 𝛼1= 𝜇/(𝜆+𝛾

1), 𝛼2= 𝜇𝛾1/𝜆(𝜆+𝛾

1),

𝛽1= 𝜆/(𝜆 + 𝛾

1) < 1, and 𝛽

2= 𝜆/(𝜆 + 𝛾

2) < 1.

Proof. From global balance we have that

(𝜆 + 𝛾1) 𝑝0,1

= 𝜇𝑝1,0

. (4)

Thus,

𝑝0,1

=𝜇

𝜆 + 𝛾1

𝑝1,0

= 𝛼1𝑝1,0

, (5)

where 𝛼1= 𝜇/(𝜆 + 𝛾

1). Similarly,

𝜆𝑝0,2

= 𝛾1𝑝0,1

, (6)

which gives

𝑝0,2

=𝛾1

𝜆𝑝0,1

= (𝛾1

𝜆)(

𝜇

𝜆 + 𝛾1

)𝑝1,0

= 𝛼2𝑝1,0

, (7)

where 𝛼2= 𝜇𝛾1/𝜆(𝜆 + 𝛾

1). Also, for 𝑛 = 0, 1, 2, . . ., we have

that𝜆𝑝𝑛,1

= (𝜆 + 𝛾1) 𝑝𝑛+1,1

,

𝜆𝑝𝑛,2

= (𝜆 + 𝛾2) 𝑝𝑛+1,2

,

(8)

which implies that

𝑝𝑛+1,1

=𝜆

𝜆 + 𝛾1

𝑝𝑛,1

𝑛 = 0, 1, 2, . . . ,

𝑝𝑛+1,2

=𝜆

𝜆 + 𝛾2

𝑝𝑛,2

𝑛 = 0, 1, 2, . . . .

(9)


Solving the above equations recursively we obtain

𝑝𝑛,1

= (𝜆

𝜆 + 𝛾1

)

𝑛

𝑝0,1

= 𝛼1(

𝜆

𝜆 + 𝛾1

)

𝑛

𝑝1,0

= 𝛼1𝛽𝑛

1𝑝1,0

𝑛 = 0, 1, 2, . . . ,

𝑝𝑛,2

= (𝜆

𝜆 + 𝛾2

)

𝑛

𝑝0,2

= 𝛼2(

𝜆

𝜆 + 𝛾2

)

𝑛

𝑝1,0

= 𝛼2𝛽𝑛

2𝑝1,0

𝑛 = 0, 1, 2, . . . ,

(10)

where 𝛽1= 𝜆/(𝜆 + 𝛾

1) < 1 and 𝛽

2= 𝜆/(𝜆 + 𝛾

2) < 1. From

local balance we obtain

𝜆𝑝𝑛,0

+ 𝜆𝑝𝑛,1

+ 𝜆𝑝𝑛,2

= 𝜇𝑝𝑛+1,0

𝑛 = 1, 2, . . . . (11)

If we define 𝜌 = 𝜆/𝜇, then for 𝑛 = 1, 2, . . ., we obtain

𝑝𝑛+1,0

= 𝜌𝑝𝑛,0

+ 𝜌𝑝𝑛,1

+ 𝜌𝑝𝑛,2

= 𝜌𝑝𝑛,0

+ 𝜌𝛼1𝛽𝑛

1𝑝1,0

+ 𝜌𝛼2𝛽𝑛

2𝑝1,0

.

(12)

Solving recursively we obtain

𝑝2,0

= 𝜌 {𝛼1𝛽1+ 𝛼2𝛽2+ 1} 𝑝

1,0

= 𝜌{𝛼1𝛽1(𝛽1− 𝜌)

𝛽1− 𝜌

+𝛼2𝛽2(𝛽2− 𝜌)

𝛽2− 𝜌

+ 𝜌0}𝑝1,0

,

𝑝3,0

= 𝜌 {𝛼1𝛽2

1+ 𝛼2𝛽2

2+ 𝜌𝛼1𝛽1+ 𝜌𝛼2𝛽2+ 𝜌} 𝑝

1,0

= 𝜌{𝛼1𝛽1(𝛽2

1− 𝜌2)

𝛽1− 𝜌

+𝛼2𝛽2(𝛽2

2− 𝜌2)

𝛽2− 𝜌

+ 𝜌}𝑝1,0

,

𝑝4,0

= 𝜌 {𝛼1𝛽3

1+ 𝛼2𝛽3

2+ 𝜌𝛼1𝛽2

1+ 𝜌𝛼2𝛽2

2

+𝜌2𝛼1𝛽1+ 𝜌2𝛼2𝛽2+ 𝜌2} 𝑝1,0

= 𝜌{𝛼1𝛽1(𝛽3

1− 𝜌3)

𝛽1− 𝜌

+𝛼2𝛽2(𝛽3

2− 𝜌3)

𝛽2− 𝜌

+ 𝜌2}𝑝1,0

.

(13)

Thus, in general we obtain

𝑝𝑛,0

= 𝜌{𝛼1𝛽1(𝛽𝑛−1

1− 𝜌𝑛−1

)

𝛽1− 𝜌

+𝛼2𝛽2(𝛽𝑛−1

2− 𝜌𝑛−1

)

𝛽2− 𝜌

+ 𝜌𝑛−2

}𝑝1,0

𝑛 = 1, 2, . . . .

(14)

From the law of total probability, we have that

1 =

∞

∑

𝑛=1

𝑝𝑛,0

+

∞

∑

𝑛=0

𝑝𝑛,1

+

∞

∑

𝑛=0

𝑝𝑛,2

= 𝑝1,0

{𝛼1

1 − 𝛽1

+𝛼2

1 − 𝛽2

+𝛼1𝛽1𝜌

(1 − 𝜌) (1 − 𝛽1)

+𝛼2𝛽2𝜌

(1 − 𝜌) (1 − 𝛽2)+

1

1 − 𝜌}

= 𝑝1,0

{((1 − 𝛽1) (1 − 𝛽

2) + 𝛼1(1 − 𝛽

2) {1 − 𝜌 (1 − 𝛽

1)}

+ 𝛼2(1 − 𝛽

1) {1 − 𝜌 (1 − 𝛽

2)})

× ((1 − 𝜌) (1 − 𝛽1) (1 − 𝛽

2))−1} .

(15)

Thus, we obtain𝑝1,0

= ((1 − 𝜌) (1 − 𝛽1) (1 − 𝛽

2))

× ((1 − 𝛽1) (1 − 𝛽

2) + 𝛼1(1 − 𝛽

2)

× {1 − 𝜌 (1 − 𝛽1)} + 𝛼

2(1 − 𝛽

1)

× {1 − 𝜌 (1 − 𝛽2)})−1,

(16)

which completes the proof.

The mean number of customers in the system is given by

𝐸 [𝑁] =

∞

∑

𝑛=1

𝑛𝑝𝑛,0

+

∞

∑

𝑛=0

𝑛𝑝𝑛,1

+

∞

∑

𝑛=0

𝑛𝑝𝑛,2

= {𝛼1𝛽1𝜌 (2 − 𝜌 − 𝛽

1)

(1 − 𝜌)2(1 − 𝛽

1)2

+𝛼2𝛽2𝜌 (2 − 𝜌 − 𝛽

2)

(1 − 𝜌)2(1 − 𝛽

2)2

+1

(1 − 𝜌)2+

𝛼1𝛽1

(1 − 𝛽1)2+

𝛼2𝛽2

(1 − 𝛽2)2}𝑝1,0

.

(17)

Finally, from Little’s formula [22] the mean time a customerspends in the system (or mean delay) is given by

𝐸 [𝑇] =𝐸 [𝑁]

𝜆. (18)

4. Computational Results

We assume that 𝜇 = 1; thus, 𝜌 = 𝜆. This implies that

𝛼1=

𝜇

𝜆 + 𝛾1

=1

𝜌 (1 + 𝛾1/𝜌)

=1

𝜌 + 𝛾1

,

𝛼2= (

𝛾1

𝜆)𝛼1=

𝛾1

𝜌2 (1 + 𝛾1/𝜌)

=𝛾1

𝜌 (𝜌 + 𝛾1),

𝛽1=

𝜆

𝜆 (1 + 𝛾1/𝜆)

=1

1 + 𝛾1/𝜌

=𝜌

𝜌 + 𝛾1

,

𝛽2=

𝜆

𝜆 (1 + 𝛾2/𝜆)

=1

1 + 𝛾2/𝜌

=𝜌

𝜌 + 𝛾2

.

(19)

Modelling and Simulation in Engineering 5

0 10.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9𝜌

0

20

40

60

80

100

120

E(T

)

𝛾1 = 1

𝛾1 = 0.5

𝛾1 = 0.25

𝛾1 = 0.1

𝛾1 = 0.05

Figure 2: Mean time in system versus offered load for 𝛾2= 1.

00

20

40

60

80

100

120

10.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

𝛾1 = 1

𝛾1 = 0.5𝛾1 = 0.1

𝛾1 = 0.25𝛾1 = 0.05

𝜌

E(T

)

Figure 3: Mean time in system versus offered load for 𝛾2= 2.

We also assume that themean duration of type 1 vacation is atleast as long as that of the type 2 duration, which means that𝛾2≥ 𝛾1. We assume that 𝛾

2= 1 and consider different values

of 𝛾1: 𝛾1= 1, 0.5, 0.25, 0.1, 0.05. Figure 2 shows the variations

of 𝐸[𝑇] with 𝜌. As the figure indicates, 𝐸[𝑇] increases as 1/𝛾1

increases (corresponding to decreasing 𝛾1).

Figure 3 shows the results for 𝛾2= 2 and 𝛾

1= 1, 0.5, 0.25,

0.1, 0.05. The figure shows the same trend observed inFigure 2.

5. Conclusion

We have considered an interesting class of multiple vacationqueueing systems inwhich two types of vacations are encoun-tered. The first type is a vacation that is taken at the endof a busy period of nonzero duration, and the second is avacation taken at the end of a busy period of zero duration,which means that no customer was served. The simple caseof M/M/1 system is considered. The results indicate that themean time a customer spends in the system, which we defineas the mean delay, is more sensitive to the mean duration ofthe first type of vacation than of the second type, which is tobe expected since the duration of the first type of vacationis assumed to be longer than that of the second type. Thisqueueing system reflects many real life experiences where

some vacations can be used for postprocessing activitieswhileothers are actual “breaks” that the server takes.

Conflict of Interests

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

Acknowledgments

Theauthors wish to acknowledge the help of NgaNguyen andOgechi Ibe in producing the graphs in Figures 2 and 3. Theyalso want to thank the reviewers for their comments on theapplication of the model to practical systems.

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