/department of mathematics and computer science 1/46

Transcription

1 LNMB Course Advanced Queueing Theory Lecture 9, April 23, 2012 Onno Boxma, Sem Borst (TU/e) sem/aqt/ /department of mathematics and computer science 1/46

2 Course overview 1. Product-form networks: Queue lengths 2. Product-form networks: Sojourn times 3. The M/G/1 queue; multi-class queues 4. Polling systems I 5. PS, symmetric disciplines, DPS, GPS, BS networks 6. Achievable delay region, delay optimization 7. Size-based scheduling, SRPT, FBPS/LAS 8. Heavy tails; impact of the service discipline 9. Polling systems II /department of mathematics and computer science 2/46

3 Introduction Optimization of polling systems has received relatively limited attention compared to analysis of polling systems for given policies and system parameters yet is broad and somewhat fragmented area /department of mathematics and computer science 3/46

4 Introduction (cont d) Will focus on: Performance objective: minimize weighted sum of mean waiting times captures both efficiency and fairness Parametrized / structured policies for arbitration between queues assuming FCFS within queues optimization of service policy (visit length): when to switch optimization of routing policy (visit frequency and order): where to switch to joint optimization of visit length and frequency Scheduling policies for priorization within queues assuming given service and routing policies /department of mathematics and computer science 4/46

5 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 performance (Lecture 6) index-type policies (e.g. cµ-rule) minimize weighted mean-waiting time objective (Lecture 7) Specific mean-waiting approximations Joint optimization of arbitration between queues (service and routing policies) and scheduling within queues /department of mathematics and computer science 5/46

6 Optimization of service policy / visit length Assume routing policy / visit order is strictly cyclic Aim to determine optimal parameters (x 1,..., x N ) for structured service policy, specifying visit lengths at various queues k i -limited τ i -limited Bernoulli(p i ) None of these service policies belongs to class of disciplines with branching property!! /department of mathematics and computer science 6/46

7 Optimization of service policy / visit length (cont d) Starting point is (approximate) expression for mean waiting time as function of design parameters (x 1,..., x N ): E{W i } F i (x 1,..., x N ) Function F i ( ) further depends on traffic-related parameters, such as arrival rate λ i, service time moments, and switch-over time moments Problem is then to minimize N i=1 c i λ i F i (x 1,..., x N ) subject to possible constraints on (x 1,..., x N ) Depending on specific form of F i (x 1,..., x N ), optimization problem may allow closed-form solution or require numerical solution procedure /department of mathematics and computer science 7/46

8 Optimization of service policy / visit length (cont d) In case of limited service policies, typical approximation is of the form F i (x 1,..., x N ) = a i + b i x i x i ρ is, 1 ρ with s representing total mean switch-over time in cycle and x i expected amount of time guaranteed per visit to Q i k i -limited: x i = k i E{B i } τ i -limited: x i = τ i Bernoulli(p i ): x i = E{B i} 1 p i Typical constraints are of the form x i > ρ is 1 ρ, i = 1,..., N, and N i=1 γ i x i G, imposing upper bound on (weighted) sum of expected visit periods, e.g., ensuring maximum expected cycle time /department of mathematics and computer science 8/46

9 Optimization of service policy / visit length (cont d) Optimization problem takes the form min N i=1 subject to x i > ρ is 1 ρ, i = 1,..., N, and c i λ i a i + b i x i x i ρ is 1 ρ N γ i x i G i=1 /department of mathematics and computer science 9/46

10 Optimization of service policy / visit length (cont d) Optimal solution is x i = ( ρ is 1 ρ + G N i=1 ) γ i ρ i s κ i 1 ρ Nj=1, κ j where κ i = c i λ i (a i + b iρ i s 1 ρ )/γ i increasing in c i, λ i, a i, E{B i } decreasing in b i, γ i Interpretation Q i needs to receive at least ρ is 1 ρ service capacity per cycle residual service capacity is divided in proportion to κ i s /department of mathematics and computer science 10/46

11 Optimization of service policy / visit length (cont d) Determining optimal service shares exactly is extremely difficult, even with zero switch-over times Resembles problem of selecting suitable weights in differentiated schedulers, 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 suitable weights for given target performance metrics remains huge challenge /department of mathematics and computer science 11/46

12 Optimization of routing policy / visit frequency and 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 order and frequency of visits to various queues random polling: probability vector (p 1,..., p N ) routing table: deterministic sequence (q 1, q 2,..., q M ) = (i 1, i 2,..., i M ) {1,..., N} M Denote by y i relative visit frequency of Q i random polling: y i = p i routing table: y i = m i M, with m i = M j=1 I {{q j =i}} /department of mathematics and computer science 12/46

13 Optimization of routing policy / visit frequency and order (cont d) Starting point is (approximate) expression for mean waiting time as function of (y 1,..., y N ): E{W i } G i (y 1,..., y N ), assuming visits to each individual queue to be evenly spaced Function G i ( ) further depends on traffic-related parameters, such as arrival rate λ i, service time moments, and switch-over time moments Problem is then to minimize N i=1 c i λ i G i (y 1,..., y N ) subject to constraint N i=1 y i = 1 Depending on specific form of G i (y 1,..., y N ), optimization problem may allow closed-form solution or require numerical solution procedure /department of mathematics and computer science 13/46

14 Optimization of routing policy / visit frequency and order (cont d) For exhaustive and gated policies, typical approximation is of the form G i (y 1,..., y N ) = a i y i N y j s j j=1 For 1-limited policies, typical approximation is of the form G i (y 1,..., y N ) = y i λ i 1 ρ b i Nj=1 y j s j N j=1 y j s j and typical constraint is of the form y i > λ i 1 ρ Nj=1 y j s j, i = 1,..., N Note that both problems are zero-degree homogeneous in (y 1,..., y N ) /department of mathematics and computer science 14/46

15 Optimization of routing policy / visit frequency and order (cont d) For gated and exhaustive policies, optimization problem takes the form min N i=1 c i λ i a i y i N y j s j j=1 subject to N i=1 y i = 1 Optimal solution is of the form y i = where κ i = c i λ i a i /s i increasing in c i, λ i, a i decreasing in s i κ i Nj=1 κ j, /department of mathematics and computer science 15/46

16 Optimization of routing policy / visit frequency and order (cont d) For 1-limited policy, optimization problem takes the form min N c i λ i i=1 y i λ i 1 ρ b i Nj=1 y j s j N j=1 y j s j subject to N i=1 y i = 1 and y i > λ is 1 ρ Nj=1 y j s j, i = 1,..., N /department of mathematics and computer science 16/46

17 Optimization of routing policy / visit frequency and order (cont d) Optimal solution is of the form yi λ i + 1 ρ N λ j s j κ i s Nj=1, i κ j j=1 where κ i = c i λ i b i /s i increasing in c i, λ i, b i decreasing in s i Interpretation Q i needs to obtain at least λ i visits (per time unit) residual visits 1 ρ N j=1 λ j s j (per time unit) in proportion to κ i /s i /department of mathematics and computer science 17/46

18 Optimization of routing policy / visit frequency and order (cont d) It remains to construct routing table based on derived visit frequencies Determine length of table, e.g., M = N i=1 m i, where ˆM y i m i N for some ˆM Determine order of visits so that visits to each individual queue are evenly spaced Even spacing is not always feasible: (m 1, m 2, m 3 ) = (1, 2, 3) (1, 2, 3, 3, 2, 3): visits to Q 2 evenly spaced, but not to Q 3 (1, 3, 2, 3, 2, 3): visits to Q 3 evenly spaced, but not to Q 2 /department of mathematics and computer science 18/46

19 Optimization of routing policy / visit frequency and 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 19/46

20 Optimization of service policy / visit length In special case of random polling with mixture of exhaustive (i e) and gated (i g) service policies at various queues and c i E{B i }, pseudo-conservation law provides exact expression for objective function: N ρ i E{W i } = ρ i=1 Ni=1 λ i E{B (2) i } 2(1 ρ) s ρi ρ p i e i s 1 ρ N i=1 ρ i p i N ρ i s i + ρ 2s i=1 N i=1 p i s (2) i In case s i s 1 and s (2) i s (2) 1, we obtain pi ρi (1 ρ i ) = j e ρ j (1 ρ j ) + i e, j g ρ j and p i = ρi j e ρ j (1 ρ j ) + j g ρ j i g /department of mathematics and computer science 20/46

21 Joint optimization of visit frequency / order and length Now suppose we wish to optimize visit lengths (x 1,..., x N ) as well as visit order and frequencies (y 1,..., y N ) Assume that condition is imposed of the form x i b i Nj=1 y j x j y i + d i, reflecting for example that visit must be sufficiently long to clear all work that arrives during intervisit period with high probability /department of mathematics and computer science 21/46

22 Joint optimization of visit frequency/order and length (cont d) As before, starting point is (approximate) expression for mean waiting time as function of (x 1,..., x N ) and (y 1,..., y N ) E{W i } H i (x 1,..., x N ; y 1,..., y N ) Function H i ( ) further depends on traffic-related parameters, such as arrival rate λ i, service time moments, and switch-over time moments Problem is then to minimize N i=1 c i λ i H i (x 1,..., x N ; y 1,..., y N ) subject to possible constraints on (x 1,..., x N ) and x i b Nj=1 i y j x j y i + d i, i = 1,..., N Depending on form of H i (x 1,..., x N ; y 1,..., y N ), optimization problem may allow closed-form solution or require numerical solution procedure /department of mathematics and computer science 22/46

23 Joint optimization of visit frequency/order and length (cont d) Typical approximation is of the form H i (x 1,..., x N ; y 1,..., y N ) a i y i N y j x j j=1 Optimization problem then takes the form min N i=1 c i λ i a i y i N y j x j j=1 subject to x i b i Nj=1 y j x j y i + d i, i = 1,..., N /department of mathematics and computer science 23/46

24 Joint optimization of visit frequency/order and length (cont d) Optimality requires latter constraint to be satisfied with equality, yielding N y j x j = j=1 Nj=1 d j y j 1 N j=1 b j, and hence optimization problem reduces to min N i=1 c i λ i a i y i N d j y j j=1 /department of mathematics and computer science 24/46

25 Joint optimization of visit frequency/order and length (cont d) Optimal solution is of the form y i = where κ i = c i λ i a i /d i κ i Nj=1 κ j, Same as optimal visit frequency for gated and exhaustive service policies /department of mathematics and computer science 25/46

26 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 (Altman, Konstantopoulos & Liu) /department of mathematics and computer science 26/46

27 Dynamic optimization (cont d) Two-class queues with set-up times / costs in symmetric scenario, optimal service policy is exhaustive, with threshold rule for switching from empty to non-empty queue (Hofri & Ross) in asymmetric scenarios, dynamic programming yields threshold rule for 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 27/46

28 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 λ i Customers at queue i have generally distributed service requirements B i Mean residual service requirement E{R Bi } = E{B2 i } 2E{B i } Traffic intensity at queue i is ρ i = λ i E{B i } Total traffic intensity is ρ = N i=1 ρ i < 1 /department of mathematics and computer science 28/46

29 Mean value analysis (cont d) Generally distributed switch-over time S i (imodn) + 1 Mean total switch-over time in cycle E{S} = N i=1 E{S i } from queue i to queue Cycle time of queue i is time between two successive arrivals of server at this queue; mean cycle time is E{C} = E{S} 1 ρ Visit time T i of queue i is service period of queue i plus preceding switchover time E{T i } = E{S i 1 } + ρ i E{C} /department of mathematics and computer science 29/46

30 Mean value analysis (cont d) Let L i denote length of queue i (excluding customer possibly in service), and let W i denote waiting time of customer at queue i Objective: Determine E{L i } and E{W i } Derive (i) arrival relation for mean waiting time and use (ii) Little s law First ordinary M/G/1 queue Arrival relation E{W} = E{L}E{B} + ρe{r B } in combination with Little s law E{L} = λe{w} yields E{W} = ρ 1 ρ E{R B} /department of mathematics and computer science 30/46

31 Mean value analysis (cont d) Now two queues Arrival relation E{W 1 } = E{L 1 }E{B 1 } + ρ 1 E{R B1 } in combination with Little s law yields E{W 1 } = 1 1 ρ 1 But what is E{R T2 }?? + E{S 2} E{C} E{R S 2 } + E{T 2} E{C} (E{R T 2 } + E{S 2 }) E{L 1 } = λ 1 E{W 1 } [ ρ 1 E{R B1 } + E{S 2} E{C} E{R S 2 } + E{T ] 2} E{C} (E{R T 2 } + E{S 2 }) /department of mathematics and computer science 31/46

32 Mean value analysis (cont d) Let L i,n denote length of queue i at arbitrary point in time during visit time of queue n Then, since each of L 2,2 customers initiates busy period, Further E{R T2 } = E{L 2,2 } E{B 2} + ρ 2E{C} E{R B2 } + E{S 1} E{R S1 } 1 ρ 2 E{T 2 } 1 ρ 2 E{T 2 } 1 ρ 2 E{L 2 } = E{T 1} E{C} E{L 2,1} + E{T 2} E{C} E{L 2,2} Finally L 2,1 is equal to number of arrivals at queue 2 during age of T 1, and age has same distribution as residual lifetime of T 1, so E{L 2,1 } = λ 2 E{R T1 } Similar equations can be obtained with roles of queues 1 and 2 interchanged /department of mathematics and computer science 32/46

33 Mean value analysis (cont d) Example: λ 1 = 0.6, λ 2 = 0.2, B 1, B 2, S 1, S 2 all exp. distr. with unit mean Solution which leads to E{L 1,2 } = 0.6E{R T2 }, E{L 2,1 } = 0.2E{R T1 }, 0.7E{L 1,1 } + 0.3E{L 1,2 } = E{R T2 }, 0.7E{L 2,1 } + 0.3E{L 2,2 } = E{R T1 }, E{R T1 } = E{L 1,1 }, E{R T2 } = E{L 2,2 } E{L 1,1 } = , E{L 1,2} = 12 5, E{L 2,1} = 82 35, E{L 2,2} = 11 5, E{R T1 } = 82 7, E{R T 2 } = 4 E{L 1 } = 3.3, E{L 2 } = 2.3, E{W 1 } = 5.5, E{W 2 } = 11.5 /department of mathematics and computer science 33/46

34 Impact of service discipline within queue So far we assumed FCFS service discipline within each queue In several computer-communication and manufacturing applications, that is not realistic We now analyze impact of service discipline within each queue (for given visit frequency and length) /department of mathematics and computer science 34/46

35 Impact of service discipline within queue (cont d) N queues, gated service discipline at Q i C i past C i res visit to Q i intervisit Consider mean waiting time E{W i (x)} of customer at Q i with service requirement x: with E{W i (x)} = E{C r i } + λ ie{c p i }E{K ip(x)} + λ i E{C r i }E{K ir(x)} K ip (x): contribution of work from type-i customer arriving earlier in cycle K ir (x): contribution of work from type-i customer arriving later in cycle /department of mathematics and computer science 35/46

36 Impact of service discipline within queue (cont d) FCFS: K ip (x) = B i, K ir (x) = 0: E{W i } = E{W i (x)} = (1 + ρ i )E{C r i } LCFS: K ip (x) = 0, K ir (x) = B i : E{W i } = E{W i (x)} = (1 + ρ i )E{C r i } PS: K ip (x) = min{b i, x}, K ir (x) = min{b i, x}: E{W i (x)} = E{C r i }(1 + 2λ ie{min{b i, x}}; E{W i } = E{C r i }(1 + 2λ ie{min{b i1, B i2 }} SPT (optimal): K ip (x) = B i I {Bi <x}, K ir (x) = B i I {Bi <x}: E{W i (x)} = E{C r i }(1 + 2λ ie{b i I {Bi <x}}); E{W i } = E{C r i }(1 + λ ie{min{b i1, B i2 }}) /department of mathematics and computer science 36/46

37 Impact of service discipline within queue (cont d) Gated: Significant gains under heavy load, but smaller than in ordinary M/G/1 queue 2 queue symmetric polling system 250 mean delay FCFS SJF load /department of mathematics and computer science 37/46

38 Impact of service discipline within queue (cont d) Exhaustive: Big gains under heavy load, matching those in ordinary M/G/1 queue 2 queue symmetric polling system 250 mean delay load /department of mathematics and computer science 38/46

39 References E. Altman, B. Gaujal, A. Hordijk (2000). Balanced sequences and optimal routing. J. ACM 47, E. Altman, P. Konstantopoulos, Z. Liu (1992). Stability, monotonicity and invariant quantities in general polling systems. Queueing Systems 11, Special Issue on Polling Models, Y. Arian, Y. Levy (1992). Algorithms for generalized round robin routing. Oper. Res. Lett. 12, J.E. Baker, I. Rubin (1987). Polling with a general-service order table. IEEE Trans. Commun. 35, J.P.C. Blanc, R.D. van der Mei (1995). Optimization of polling systems with Bernoulli schedules. Perf. Eval. 21, S.C. Borst, O.J. Boxma, J.H.A. Harink, G.B. Huitema (1994). Optimization of fixed time polling schemes. Telecommunication Systems 3, S.C. Borst, O.J. Boxma, H. Levy (1995). The use of service limits for efficient operation of multi-station single-medium communication systems. IEEE/ACM Trans. Netw. 3, /department of mathematics and computer science 39/46

40 References (cont d) S.C. Borst, K.G. Ramakrishnan (1999). Optimization of template-driven scheduling mechanisms: regularity measures and computational techniques. J. Scheduling 2, O.J. Boxma (1991). Analysis and optimization of polling systems. In: Queueing, Performance and Control in ATM, eds. J.W. Cohen, C.D. Pack, O.J. Boxma, J. Bruin, B. Fralix (2009). Sojourn times in polling systems with various service disciplines. Perf. Eval 66 (11), O.J. Boxma, W.P. Groenendijk, J.A. Weststrate (1990). A pseudoconservation law for service systems with a polling table. IEEE Trans. Commun. 38, O.J. Boxma, H. Levy, J.A. Weststrate (1990). Optimization of polling systems. In: Proc. Performance 90, eds. P.J.B. King, I. Mitrani, R.J. Pooley, O.J. Boxma, H. Levy, J.A. Weststrate (1991). Efficient visit frequencies for polling tables: minimization of waiting cost. Queueing Systems 9, /department of mathematics and computer science 40/46

41 References (cont d) O.J. Boxma, H. Levy, J.A. Weststrate (1993). Efficient visit orders for polling systems. Perf. Eval. 18, O.J. Boxma, B.W. Meister (1987). Waiting-time approximations for cyclicservice systems with switchover times. Perf. Eval. 7, O.J. Boxma, B.W. Meister (1987). Waiting-time approximations in multiqueue systems with cyclic-service. Perf. Eval. 7, O.J. Boxma, J.A. Weststrate (1989). Waiting times in polling systems with Markovian server routing. In: Messung, Modellierung und Bewertung von Rechensystemen und Netzen, eds. G. Stiege, J.S. Lie (Springer, Berlin), S. Browne, G. Weiss (1992). Dynamic priority rules when polling with multiple parallel servers. Oper. Res. Lett. 12, S. Browne, U. Yechiali (1989). Dynamic priority rules for cyclic-type queues. Adv. Appl. Prob. 21, W. Bux, H.L. Truong (1983). Mean-delay approximations for cyclic-service queueing systems. Perf. Eval. 3, /department of mathematics and computer science 41/46

42 References (cont d) C. Buyukkoc, P. Varaiya, J. Walrand (1985). The cµ rule revisited. Adv. Appl. Prob. 17, K.C. Chang, D. Sandhu (1992). Mean waiting time approximations in cyclicservice systems with exhaustive limited service policy. Perf. Eval. 15, D.E. Everitt (1986). A conservation-type law for the token ring with limited service. Br. Telecom Techn. J. 4, D.E. Everitt (1986). Simple approximations for token rings. IEEE Trans. Commun. 34, D.E. Everitt (1989). An approximation procedure for cyclic service queues with limited service. In: Performance of Distributed and Parallel Systems, eds. T. Hasegawa, H. Takagi, Y. Takahashi, O. Fabian, H. Levy (1994). Pseudo-cyclic policies for multi-queue single server systems. Ann. Oper. Res. 48, Special Issue on Queueing Networks, ed. N.M. van Dijk, S.W. Fuhrmann, Y.T. Wang (1988). Analysis of cyclic service systems with limited service: bounds and approximations. Perf. Eval. 9, /department of mathematics and computer science 42/46

43 References (cont d) B. Gaujal, A. Hordijk, D.A. van der Laan (2007). On the optimal open-loop control policy for deterministic and exponential polling systems. Prob. Eng Inf. Sc. 21, W.P. Groenendijk (1990). Conservation Laws in Polling Systems. PhD Thesis University of Utrecht. W.P. Groenendijk (1989). Waiting-time approximations for cyclic service systems with mixed service strategies. In: Teletraffic Science for New Cost- Effective Systems, Networks and Services, ITC 12, ed. M. Bonatti, B. Hajek (1985). Extremal splittings of point processes. Math. Oper. Res. 10, M. Hofri, Z. Rosberg (1987). Packet delay under the Golden Ratio weighted TDM policy in a multiple-access channel. IEEE Trans. Inform. Theory 33, M. Hofri, K.W. Ross (1987). On the optimal control of two queues with server set-up times and its analysis. SIAM J. Comput. 16, /department of mathematics and computer science 43/46

44 References (cont d) A. Itai, Z. Rosberg (1984). A Golden Ratio control policy for a multipleaccess channel. IEEE Trans. Autom. Control 29, G.M. Koole (1997). Assigning a single server to inhomogeneous queues with switching costs. Th. Comp. Sc. 182, J.B. Kruskal (1969). Work-scheduling algorithms: a non-probabilistic queueing study (with possible applications to No. 1 ESS). Bell Syst. Techn. J. 48, D.A. van der Laan (2003). The Structure and Performance of Optimal Routing Sequences. PhD Thesis Leiden University. H. Levy (1988). Optimization of polling systems: the fractional exhaustive service method. Report Tel Aviv University, Tel Aviv. H. Levy, M. Sidi (1990). Polling systems: applications, modelling and optimization. IEEE Trans. Commun. 38, H. Levy, M. Sidi, O.J. Boxma (1990). Dominance relations in polling systems. Queueing Systems 6, /department of mathematics and computer science 44/46

45 References (cont d) Z. Liu, P. Nain, D. Towsley (1992). On optimal polling policies. Queueing Systems 11, Special Issue on Polling Models, D.M. Markowitz, M.I. Reiman, L.M. Wein (2000). The stochastic economic lot scheduling problem: heavy-traffic analysis of dynamic cyclic policies. Oper. Res. 48, D.M. Markowitz, L.M. Wein (2001). Heavy-traffic analysis of dynamic cyclic policies: A unified treatment of the single machine scheduling problem. Oper. Res. 49, I. Meilijson, U. Yechiali (1977). On optimal right-of-way policies at a singleserver station when insertion of idle times is permitted. Stoch. Proc. Appl. 6, S.S. Panwar, T.K. Philips, M.-S. Chen (1992). Golden Ratio scheduling for flow control with low buffer requirements. IEEE Trans. Commun. 40, M.I. Reiman, L.M. Wein (1998). Dynamic scheduling of a two-class queue with setups. Oper. Res. 46, /department of mathematics and computer science 45/46

46 References (cont d) L.D. Servi (1986). Average delay approximation of M / G / 1 cyclic service queue with Bernoulli schedule. IEEE J. Sel. Areas Commun. 4, J.A. Weststrate (1992). Analysis and Optimization of Polling Models. PhD Thesis University of Tilburg. A.C. Wierman, E.M.M. Winands, O.J. Boxma (2007). Scheduling in polling systems. Perf. Eval 64 (9 12), E.M.M. Winands (2007). Polling, Production and Priorities. PhD Thesis Eindhoven University of Technology. E.M.M. Winands, I.J.B.F. Adan, G.J. van Houtum (2006). Mean value analysis for polling systems. Queueing Systems 54, U. Yechiali (1991). Optimal dynamic control of polling systems. In: Queueing, Performance and Control in ATM, eds. J.W. Cohen, C.D. Pack, /department of mathematics and computer science 46/46