lnmb course advanced queueing theorysem/aqt/lecture23042012.pdf · department of mathematics and...
TRANSCRIPT
/ department of mathematics and computer science 1/46
LNMB Course
Advanced Queueing Theory
Lecture 9, April 23, 2012
Onno Boxma, Sem Borst (TU/e)
http://www.win.tue.nl/˜sem/AQT/
/ department of mathematics and computer science 2/46
Course overview
1. Product-form networks: Queue lengths2. Product-form networks: Sojourn times3. The M/G/1 queue; multi-class queues4. Polling systems I5. PS, symmetric disciplines, DPS, GPS, BS networks6. Achievable delay region, delay optimization7. Size-based scheduling, SRPT, FBPS/LAS8. Heavy tails; impact of the service discipline9. Polling systems II
/ department of mathematics and computer science 3/46
Introduction
Optimization of polling systems
• has received relatively limited attention compared to analysis of pollingsystems for given policies and system parameters
• yet is broad and somewhat fragmented area
/ department of mathematics and computer science 4/46
Introduction (cont’d)
Will focus on:
• Performance objective: minimize weighted sum of mean waiting timescaptures both efficiency and fairness
• Parametrized / structured policies for arbitration between queuesassuming FCFS within queues
– optimization of service policy (visit length): when to switch
– optimization of routing policy (visit frequency and order): where toswitch to
– joint optimization of visit length and frequency
• Scheduling policies for priorization within queuesassuming given service and routing policies
/ department of mathematics and computer science 5/46
Introduction (cont’d)
Will not consider:
• Polling systems with zero switch-over times
– correspond to ‘ordinary’ single-server multi-class systems
– polyhedral characterization of achievable mean-waiting time perfor-mance (Lecture 6)
– index-type policies (e.g. cµ-rule) minimize weighted mean-waitingtime objective (Lecture 7)
• Specific mean-waiting approximations
• Joint optimization of arbitration between queues (service and routingpolicies) and scheduling within queues
/ department of mathematics and computer science 6/46
Optimization of service policy / visit length
Assume routing policy / visit order is strictly cyclic
Aim to determine optimal parameters (x1, . . . , xN ) for structured servicepolicy, specifying visit lengths at various queues
• ki -limited
• τi -limited
• Bernoulli(pi )
None of these service policies belongs to class of disciplines with branchingproperty!!
/ department of mathematics and computer science 7/46
Optimization of service policy / visit length (cont’d)
Starting point is (approximate) expression for mean waiting time as functionof design parameters (x1, . . . , xN ):
E{Wi} ≈ Fi(x1, . . . , xN )
Function Fi(·) further depends on traffic-related parameters, such as arrivalrate λi , service time moments, and switch-over time moments
Problem is then to minimize∑N
i=1 ciλi Fi(x1, . . . , xN )
subject to possible constraints on (x1, . . . , xN )
Depending on specific form of Fi(x1, . . . , xN ), optimization problem mayallow closed-form solution or require numerical solution procedure
/ department of mathematics and computer science 8/46
Optimization of service policy / visit length (cont’d)
In case of ‘limited’ service policies, typical approximation is of the form
Fi(x1, . . . , xN ) =ai + bi xi
xi −ρi s
1−ρ,
with s representing total mean switch-over time in cycle and xi expectedamount of time guaranteed per visit to Qi
• ki -limited: xi = kiE{Bi}
• τi -limited: xi = τi
• Bernoulli(pi ): xi =E{Bi }1−pi
Typical constraints are of the form xi >ρi s
1−ρ , i = 1, . . . , N , and∑N
i=1 γi xi ≤
G, imposing upper bound on (weighted) sum of expected visit periods, e.g.,ensuring maximum expected cycle time
/ department of mathematics and computer science 9/46
Optimization of service policy / visit length (cont’d)
Optimization problem takes the form
minN∑
i=1
ciλiai + bi xi
xi −ρi s
1−ρ
subject to xi >ρi s
1−ρ , i = 1, . . . , N , and
N∑i=1
γi xi ≤ G
/ department of mathematics and computer science 10/46
Optimization of service policy / visit length (cont’d)
Optimal solution is
x∗i =ρis
1− ρ+
(G −
N∑i=1
γiρis1− ρ
)κi∑N
j=1 κ j,
where
κi =
√ciλi(ai +
biρis1− ρ
)/γi
• increasing in ci , λi , ai , E{Bi}
• decreasing in bi , γi
Interpretation
• Qi needs to receive at least ρi s1−ρ service capacity per cycle
• residual service capacity is divided in proportion to κi ’s
/ department of mathematics and computer science 11/46
Optimization of service policy / visit length (cont’d)
Determining optimal service shares exactly is extremely difficult, even withzero switch-over times
• Resembles problem of selecting suitable weights in differentiated sched-ulers, such as
– Weighted Round Robin (WRR)
– Weighted Fair Queueing (WFQ)
– Generalized Processor Sharing (GPS)
– Discriminatory Processor Sharing (DPS)
• While such schedulers are widely believed to be useful, setting suitableweights for given target performance metrics remains huge challenge
/ department of mathematics and computer science 12/46
Optimization of routing policy / visit frequencyand order
Now suppose service policy at each of queues is given, e.g., exhaustive,gated, or 1-limited
Aim to determine optimal parameters for routing policy, specifying orderand frequency of visits to various queues
• random polling: probability vector (p1, . . . , pN )
• routing table: deterministic sequence (q1, q2, . . . , qM) =
(i1, i2, . . . , iM) ∈ {1, . . . , N }M
Denote by yi relative visit frequency of Qi
• random polling: yi = pi
• routing table: yi =miM , with mi =
∑Mj=1 I{{q j=i}}
/ department of mathematics and computer science 13/46
Optimization of routing policy / visit frequency and order (cont’d)
Starting point is (approximate) expression for mean waiting time as functionof (y1, . . . , yN ):
E{Wi} ≈ Gi(y1, . . . , yN ),
assuming visits to each individual queue to be evenly spaced
Function Gi(·) further depends on traffic-related parameters, such as arrivalrate λi , service time moments, and switch-over time moments
Problem is then to minimize∑N
i=1 ciλi Gi(y1, . . . , yN )
subject to constraint∑N
i=1 yi = 1
Depending on specific form of Gi(y1, . . . , yN ), optimization problem mayallow closed-form solution or require numerical solution procedure
/ department of mathematics and computer science 14/46
Optimization of routing policy / visit frequencyand order (cont’d)
For exhaustive and gated policies, typical approximation is of the form
Gi(y1, . . . , yN ) =ai
yi
N∑j=1
y js j
For 1-limited policies, typical approximation is of the form
Gi(y1, . . . , yN ) =bi
yi −λi
1−ρ∑N
j=1 y js j
N∑j=1
y js j
and typical constraint is of the form yi >λi
1−ρ∑N
j=1 y js j , i = 1, . . . , N
Note that both problems are zero-degree homogeneous in (y1, . . . , yN )
/ department of mathematics and computer science 15/46
Optimization of routing policy / visit frequencyand order (cont’d)
For gated and exhaustive policies, optimization problem takes the form
minN∑
i=1
ciλiai
yi
N∑j=1
y js j
subject to∑N
i=1 yi = 1
Optimal solution is of the form y∗i =κi∑N
j=1 κ j,
where κi =√
ciλiai/si
• increasing in ci , λi , ai
• decreasing in si
/ department of mathematics and computer science 16/46
Optimization of routing policy / visit frequencyand order (cont’d)
For 1-limited policy, optimization problem takes the form
minN∑
i=1
ciλibi
yi −λi
1−ρ∑N
j=1 y js j
N∑j=1
y js j
subject to∑N
i=1 yi = 1 and yi >λi s
1−ρ∑N
j=1 y js j , i = 1, . . . , N
/ department of mathematics and computer science 17/46
Optimization of routing policy / visit frequencyand order (cont’d)
Optimal solution is of the form
y∗i ∼ λi +
1− ρ −N∑
j=1
λ js j
κi
si∑N
j=1 κ j,
where κi =√
ciλibi/si
• increasing in ci , λi , bi
• decreasing in si
Interpretation
• Qi needs to obtain at least λi visits (per time unit)
• residual visits 1− ρ −∑N
j=1 λ js j (per time unit) in proportion to κi/si
/ department of mathematics and computer science 18/46
Optimization of routing policy / visit frequencyand order (cont’d)
It remains to construct routing table based on derived visit frequencies
• Determine length of table, e.g., M =∑N
i=1 mi , where M̂ yi ≈ mi ∈ N forsome M̂
• Determine order of visits so that visits to each individual queue areevenly spaced
Even spacing is not always feasible: (m1,m2,m3) = (1, 2, 3)
• (1, 2, 3, 3, 2, 3): visits to Q2 evenly spaced, but not to Q3
• (1, 3, 2, 3, 2, 3): visits to Q3 evenly spaced, but not to Q2
/ department of mathematics and computer science 19/46
Optimization of routing policy / visit frequencyand order (cont’d)
Approaches for spacing visits ‘as evenly as possible’
• Golden Ratio rule (Hofri & Rosberg, Panwar)
• balanced sequences (Altman, Gaujal, Hordijk, Van der Laan)
• interleaving of optimal splitting sequences (Arian & Y. Levy, B.)
• quadratic programming techniques (B. & Ramakrishnan)
/ department of mathematics and computer science 20/46
Optimization of service policy / visit lengthIn special case of random polling with mixture of exhaustive (i ∈ e) andgated (i ∈ g) service policies at various queues and ci ≡ E{Bi},pseudo-conservation law provides exact expression for objective function:
N∑i=1
ρiE{Wi } = ρ
∑Ni=1 λiE{B
(2)i }
2(1− ρ)−
s1− ρ
∑i∈e
ρ2i
pi+
s1− ρ
N∑i=1
ρi
pi−
N∑i=1
ρi si +ρ
2s
N∑i=1
pi s(2)i
In case si ≡ s1 and s(2)i ≡ s(2)1 , we obtain
p∗i =√ρi(1− ρi)∑
j∈e√ρ j(1− ρ j)+
∑j∈g√ρ j
i ∈ e,
and
p∗i =√ρi∑
j∈e√ρ j(1− ρ j)+
∑j∈g√ρ j
i ∈ g
/ department of mathematics and computer science 21/46
Joint optimization of visit frequency / order and length
Now suppose we wish to optimize visit lengths (x1, . . . , xN ) as well as visitorder and frequencies (y1, . . . , yN )
Assume that condition is imposed of the form
xi ≥bi∑N
j=1 y j x j
yi+ di ,
reflecting for example that visit must be sufficiently long to clear all workthat arrives during intervisit period with high probability
/ department of mathematics and computer science 22/46
Joint optimization of visit frequency/order andlength (cont’d)
As before, starting point is (approximate) expression for mean waiting timeas function of (x1, . . . , xN ) and (y1, . . . , yN )
E{Wi} ≈ Hi(x1, . . . , xN ; y1, . . . , yN )
Function Hi(·) further depends on traffic-related parameters, such as arrivalrate λi , service time moments, and switch-over time moments
Problem is then to minimize∑N
i=1 ciλi Hi(x1, . . . , xN ; y1, . . . , yN )
subject to possible constraints on (x1, . . . , xN ) and xi ≥bi∑N
j=1 y j x jyi
+ di ,i = 1, . . . , N
Depending on form of Hi(x1, . . . , xN ; y1, . . . , yN ), optimization problemmay allow closed-form solution or require numerical solution procedure
/ department of mathematics and computer science 23/46
Joint optimization of visit frequency/order andlength (cont’d)
Typical approximation is of the form
Hi(x1, . . . , xN ; y1, . . . , yN ) ≈ai
yi
N∑j=1
y j x j
Optimization problem then takes the form
minN∑
i=1
ciλiai
yi
N∑j=1
y j x j
subject to xi ≥bi∑N
j=1 y j x jyi
+ di , i = 1, . . . , N
/ department of mathematics and computer science 24/46
Joint optimization of visit frequency/order andlength (cont’d)
Optimality requires latter constraint to be satisfied with equality, yielding
N∑j=1
y j x j =
∑Nj=1 d j y j
1−∑N
j=1 b j,
and hence optimization problem reduces to
minN∑
i=1
ciλiai
yi
N∑j=1
d j y j
/ department of mathematics and computer science 25/46
Joint optimization of visit frequency/order andlength (cont’d)
Optimal solution is of the form y∗i =κi∑N
j=1 κ j,
where κi =√
ciλiai/di
Same as optimal visit frequency for gated and exhaustive service policies
/ department of mathematics and computer science 26/46
Dynamic optimization
We have focused on ‘static’ optimization of parametrized/structured policies
Dynamic policies have received relatively limited attention
• optimization of visit order from cycle to cycle (Browne, Weiss, Yechiali)
• stochastic optimality results for total workload and queue length (Liu,Nain & Towsley)
• dominance relationships for total workload (H. Levy, Sidi & Boxma)
• stochastic monotonicity results for queue lengths at visit epochs (Alt-man, Konstantopoulos & Liu)
/ department of mathematics and computer science 27/46
Dynamic optimization (cont’d)
Two-class queues with set-up times / costs
• in symmetric scenario, optimal service policy is exhaustive, with thresh-old rule for switching from empty to non-empty queue (Hofri & Ross)
• in asymmetric scenarios, dynamic programming yields threshold rulefor switching from ‘cheap’ to ‘expensive’ queue (Koole)
• asymptotically optimal policies in heavy-traffic regime (Reiman & Wein)
• heavy-traffic analysis of dynamic cyclic policies (Markowitz, Reiman &Wein)
Spatial settings
• vehicle routing strategies
• message gathering strategies in wireless networks
/ department of mathematics and computer science 28/46
Mean value analysis
Approach for determining mean queue lengths and waiting times based on
• PASTA property: Poisson Arrivals See Time Averages
• Little’s law: E{L} = λE{W}
N queues 1, 2, . . . , N , served by single server in fixed cyclic order
• Exhaustive service discipline and FCFS within each queue
• Customers arrive at queue i as Poisson process of rate λi
• Customers at queue i have generally distributed service requirements Bi
• Mean residual service requirement E{RBi } =E{B2
i }
2E{Bi }
• Traffic intensity at queue i is ρi = λiE{Bi}
• Total traffic intensity is ρ =∑N
i=1 ρi < 1
/ department of mathematics and computer science 29/46
Mean value analysis (cont’d)
• Generally distributed switch-over time Si from queue i to queue(imodN )+ 1
• Mean total switch-over time in cycle E{S} =∑N
i=1 E{Si}
• Cycle time of queue i is time between two successive arrivals of server atthis queue; mean cycle time is
E{C} = E{S}1− ρ
• Visit time Ti of queue i is service period of queue i plus precedingswitchover time
E{Ti} = E{Si−1} + ρiE{C}
/ department of mathematics and computer science 30/46
Mean value analysis (cont’d)
Let Li denote length of queue i (excluding customer possibly in service),and let Wi denote waiting time of customer at queue i
Objective: Determine E{Li} and E{Wi}
Derive (i) arrival relation for mean waiting time and use (ii) Little’s law
First ordinary M/G/1 queueArrival relation
E{W} = E{L}E{B} + ρE{RB}
in combination with Little’s law
E{L} = λE{W}
yields
E{W} = ρ
1− ρE{RB}
/ department of mathematics and computer science 31/46
Mean value analysis (cont’d)
Now two queuesArrival relation
E{W1} = E{L1}E{B1} + ρ1E{RB1}
+E{S2}
E{C} E{RS2} +E{T2}
E{C} (E{RT2} + E{S2})
in combination with Little’s law
E{L1} = λ1E{W1}
yields
E{W1} =1
1− ρ1
[ρ1E{RB1} +
E{S2}
E{C} E{RS2} +E{T2}
E{C} (E{RT2} + E{S2})
]But what is E{RT2}??
/ department of mathematics and computer science 32/46
Mean value analysis (cont’d)
Let Li,n denote length of queue i at arbitrary point in time during visit timeof queue n
Then, since each of L2,2 customers initiates busy period,
E{RT2} = E{L2,2}E{B2}
1− ρ2+ρ2E{C}E{T2}
E{RB2}
1− ρ2+
E{S1}
E{T2}
E{RS1}
1− ρ2
Further
E{L2} =E{T1}
E{C} E{L2,1} +E{T2}
E{C} E{L2,2}
Finally L2,1 is equal to number of arrivals at queue 2 during age of T1, andage has same distribution as residual lifetime of T1, so
E{L2,1} = λ2E{RT1}
Similar equations can be obtained with roles of queues 1 and 2 interchanged
/ department of mathematics and computer science 33/46
Mean value analysis (cont’d)
Example: λ1 = 0.6, λ2 = 0.2, B1, B2, S1, S2 all exp. distr. with unit mean
E{L1,2} = 0.6E{RT2},
E{L2,1} = 0.2E{RT1},
0.7E{L1,1} + 0.3E{L1,2} = 1.5+ 0.45E{RT2},
0.7E{L2,1} + 0.3E{L2,2} = 0.25+ 0.175E{RT1},
E{RT1} = 2.5+ 2.5E{L1,1},
E{RT2} = 1.25+ 1.25E{L2,2}
Solution
E{L1,1} =12935 , E{L1,2} =
125 , E{L2,1} =
8235, E{L2,2} =
115 ,
E{RT1} =827 , E{RT2} = 4
which leads to
E{L1} = 3.3, E{L2} = 2.3, E{W1} = 5.5, E{W2} = 11.5
/ department of mathematics and computer science 34/46
Impact of service discipline within queue
So far we assumed FCFS service discipline within each queue
In several computer-communication and manufacturing applications, thatis not realistic
We now analyze impact of service discipline within each queue(for given visit frequency and length)
/ department of mathematics and computer science 35/46
Impact of service discipline within queue (cont’d)
N queues, gated service discipline at Qi
visitto Qi
intervisit
Ci
pastCi
res
Consider mean waiting time E{Wi(x)} of customer at Qi with service re-quirement x :
E{Wi(x)} = E{Cri } + λiE{Cp
i }E{Ki p(x)} + λiE{Cri }E{Kir(x)}
with
• Ki p(x): contribution of work from type-i customer arriving earlier incycle
• Kir(x): contribution of work from type-i customer arriving later in cycle
/ department of mathematics and computer science 36/46
Impact of service discipline within queue (cont’d)
• FCFS: Ki p(x) = Bi , Kir(x) = 0:
E{Wi} = E{Wi(x)} = (1+ ρi)E{Cri }
• LCFS: Ki p(x) = 0, Kir(x) = Bi :
E{Wi} = E{Wi(x)} = (1+ ρi)E{Cri }
• PS: Ki p(x) = min{Bi , x}, Kir(x) = min{Bi , x}:
E{Wi(x)} = E{Cri }(1+ 2λiE{min{Bi , x}};
E{Wi} = E{Cri }(1+ 2λiE{min{Bi1,Bi2}}
• SPT (optimal): Ki p(x) = BiI{Bi<x}, Kir(x) = BiI{Bi<x}:
E{Wi(x)} = E{Cri }(1+ 2λiE{BiI{Bi<x}});
E{Wi} = E{Cri }(1+ λiE{min{Bi1,Bi2}})
/ department of mathematics and computer science 37/46
Impact of service discipline within queue(cont’d)
Gated:Significant gains under heavy load, but smaller than in ordinary M/G/1queue
250
200
150
100
50
0.6 0.7 0.8 0.9 1
SJF
FCFS
2 queue symmetric polling system
load
mean d
ela
y
/ department of mathematics and computer science 38/46
Impact of service discipline within queue(cont’d)
Exhaustive:Big gains under heavy load, matching those in ordinary M/G/1 queue
250
200
150
100
50
0.6 0.7 0.8 0.9 1
2 queue symmetric polling system
load
mean d
ela
y
/ department of mathematics and computer science 39/46
References
E. Altman, B. Gaujal, A. Hordijk (2000). Balanced sequences and optimalrouting. J. ACM 47, 752–775.E. Altman, P. Konstantopoulos, Z. Liu (1992). Stability, monotonicity andinvariant quantities in general polling systems. Queueing Systems 11, SpecialIssue on Polling Models, 35–57.Y. Arian, Y. Levy (1992). Algorithms for generalized round robin routing.Oper. Res. Lett. 12, 313–319.J.E. Baker, I. Rubin (1987). Polling with a general-service order table. IEEETrans. Commun. 35, 283–288.J.P.C. Blanc, R.D. van der Mei (1995). Optimization of polling systems withBernoulli schedules. Perf. Eval. 21, 139–158.S.C. Borst, O.J. Boxma, J.H.A. Harink, G.B. Huitema (1994). Optimizationof fixed time polling schemes. Telecommunication Systems 3, 31–59.S.C. Borst, O.J. Boxma, H. Levy (1995). The use of service limits for ef-ficient operation of multi-station single-medium communication systems.IEEE/ACM Trans. Netw. 3, 602–612.
/ department of mathematics and computer science 40/46
References (cont’d)
S.C. Borst, K.G. Ramakrishnan (1999). Optimization of template-drivenscheduling mechanisms: regularity measures and computational tech-niques. J. Scheduling 2, 19–33.O.J. Boxma (1991). Analysis and optimization of polling systems. In: Queue-ing, Performance and Control in ATM, eds. J.W. Cohen, C.D. Pack, 173–183.O.J. Boxma, J. Bruin, B. Fralix (2009). Sojourn times in polling systemswith various service disciplines. Perf. Eval 66 (11), 621–639.O.J. Boxma, W.P. Groenendijk, J.A. Weststrate (1990). A pseudo-conservation law for service systems with a polling table. IEEE Trans. Com-mun. 38, 1865–1870.O.J. Boxma, H. Levy, J.A. Weststrate (1990). Optimization of polling sys-tems. In: Proc. Performance ’90, eds. P.J.B. King, I. Mitrani, R.J. Pooley,349–361.O.J. Boxma, H. Levy, J.A. Weststrate (1991). Efficient visit frequencies forpolling tables: minimization of waiting cost. Queueing Systems 9, 133–162.
/ department of mathematics and computer science 41/46
References (cont’d)
O.J. Boxma, H. Levy, J.A. Weststrate (1993). Efficient visit orders for pollingsystems. Perf. Eval. 18, 103–123.O.J. Boxma, B.W. Meister (1987). Waiting-time approximations for cyclic-service systems with switchover times. Perf. Eval. 7, 299–308.O.J. Boxma, B.W. Meister (1987). Waiting-time approximations in multi-queue systems with cyclic-service. Perf. Eval. 7, 59–70.O.J. Boxma, J.A. Weststrate (1989). Waiting times in polling systems withMarkovian server routing. In: Messung, Modellierung und Bewertung vonRechensystemen und Netzen, eds. G. Stiege, J.S. Lie (Springer, Berlin), 89–104.S. Browne, G. Weiss (1992). Dynamic priority rules when polling with mul-tiple parallel servers. Oper. Res. Lett. 12, 129–137.S. Browne, U. Yechiali (1989). Dynamic priority rules for cyclic-type queues.Adv. Appl. Prob. 21, 432-450.W. Bux, H.L. Truong (1983). Mean-delay approximations for cyclic-servicequeueing systems. Perf. Eval. 3, 187–196.
/ department of mathematics and computer science 42/46
References (cont’d)
C. Buyukkoc, P. Varaiya, J. Walrand (1985). The cµ rule revisited. Adv. Appl.Prob. 17, 237–238.K.C. Chang, D. Sandhu (1992). Mean waiting time approximations in cyclic-service systems with exhaustive limited service policy. Perf. Eval. 15, 21–40.D.E. Everitt (1986). A conservation-type law for the token ring with limitedservice. Br. Telecom Techn. J. 4, 51–61.D.E. Everitt (1986). Simple approximations for token rings. IEEE Trans.Commun. 34, 719–721.D.E. Everitt (1989). An approximation procedure for cyclic service queueswith limited service. In: Performance of Distributed and Parallel Systems, eds.T. Hasegawa, H. Takagi, Y. Takahashi, 141–156.O. Fabian, H. Levy (1994). Pseudo-cyclic policies for multi-queue singleserver systems. Ann. Oper. Res. 48, Special Issue on Queueing Networks, ed.N.M. van Dijk, 127–152.S.W. Fuhrmann, Y.T. Wang (1988). Analysis of cyclic service systems withlimited service: bounds and approximations. Perf. Eval. 9, 35–54.
/ department of mathematics and computer science 43/46
References (cont’d)
B. Gaujal, A. Hordijk, D.A. van der Laan (2007). On the optimal open-loopcontrol policy for deterministic and exponential polling systems. Prob. EngInf. Sc. 21, 157–187.W.P. Groenendijk (1990). Conservation Laws in Polling Systems. PhD ThesisUniversity of Utrecht.W.P. Groenendijk (1989). Waiting-time approximations for cyclic servicesystems with mixed service strategies. In: Teletraffic Science for New Cost-Effective Systems, Networks and Services, ITC 12, ed. M. Bonatti, 1434–1441.B. Hajek (1985). Extremal splittings of point processes. Math. Oper. Res. 10,543–556.M. Hofri, Z. Rosberg (1987). Packet delay under the Golden Ratio weightedTDM policy in a multiple-access channel. IEEE Trans. Inform. Theory 33,341–349.M. Hofri, K.W. Ross (1987). On the optimal control of two queues withserver set-up times and its analysis. SIAM J. Comput. 16, 399–420.
/ department of mathematics and computer science 44/46
References (cont’d)
A. Itai, Z. Rosberg (1984). A Golden Ratio control policy for a multiple-access channel. IEEE Trans. Autom. Control 29, 712–718.G.M. Koole (1997). Assigning a single server to inhomogeneous queueswith switching costs. Th. Comp. Sc. 182, 203–216.J.B. Kruskal (1969). Work-scheduling algorithms: a non-probabilisticqueueing study (with possible applications to No. 1 ESS). Bell Syst. Techn.J. 48, 2963–2974.D.A. van der Laan (2003). The Structure and Performance of Optimal RoutingSequences. PhD Thesis Leiden University.H. Levy (1988). Optimization of polling systems: the fractional exhaustiveservice method. Report Tel Aviv University, Tel Aviv.H. Levy, M. Sidi (1990). Polling systems: applications, modelling and opti-mization. IEEE Trans. Commun. 38, 1750–1760.H. Levy, M. Sidi, O.J. Boxma (1990). Dominance relations in polling sys-tems. Queueing Systems 6, 155–171.
/ department of mathematics and computer science 45/46
References (cont’d)
Z. Liu, P. Nain, D. Towsley (1992). On optimal polling policies. QueueingSystems 11, Special Issue on Polling Models, 59–83.D.M. Markowitz, M.I. Reiman, L.M. Wein (2000). The stochastic economiclot scheduling problem: heavy-traffic analysis of dynamic cyclic policies.Oper. Res. 48, 136–154.D.M. Markowitz, L.M. Wein (2001). Heavy-traffic analysis of dynamic cyclicpolicies: A unified treatment of the single machine scheduling problem.Oper. Res. 49, 246–270.I. Meilijson, U. Yechiali (1977). On optimal right-of-way policies at a single-server station when insertion of idle times is permitted. Stoch. Proc. Appl. 6,25–32.S.S. Panwar, T.K. Philips, M.-S. Chen (1992). Golden Ratio scheduling forflow control with low buffer requirements. IEEE Trans. Commun. 40, 765–772.M.I. Reiman, L.M. Wein (1998). Dynamic scheduling of a two-class queuewith setups. Oper. Res. 46, 532–547.
/ department of mathematics and computer science 46/46
References (cont’d)
L.D. Servi (1986). Average delay approximation of M / G / 1 cyclic servicequeue with Bernoulli schedule. IEEE J. Sel. Areas Commun. 4, 813–822.J.A. Weststrate (1992). Analysis and Optimization of Polling Models. PhDThesis University of Tilburg.A.C. Wierman, E.M.M. Winands, O.J. Boxma (2007). Scheduling in pollingsystems. Perf. Eval 64 (9–12), 1009–1028.E.M.M. Winands (2007). Polling, Production and Priorities. PhD Thesis Eind-hoven University of Technology.E.M.M. Winands, I.J.B.F. Adan, G.J. van Houtum (2006). Mean value anal-ysis for polling systems. Queueing Systems 54, 35–44.U. Yechiali (1991). Optimal dynamic control of polling systems. In: Queue-ing, Performance and Control in ATM, eds. J.W. Cohen, C.D. Pack, 205–217.