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Queueing Systems

, Volume 90, Issue 1–2, pp 1–33 | Cite as

Perfect sampling of GI/GI/c queues

  • Jose Blanchet
  • Jing Dong
  • Yanan Pei
Article
  • 109 Downloads

Abstract

We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backward in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.

Keywords

Perfect sampling FCFS multi-server queue Dominated coupling from the past Random walks 

Mathematics Subject Classification

60K25 

References

  1. 1.
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, Berlin (2003)Google Scholar
  2. 2.
    Asmussen, S., Glynn, P., Thorisson, H.: Stationarity detection in the initial transient problem. ACM Trans. Model. Comput. Simul. (TOMACS) 2(2), 130–157 (1992)CrossRefGoogle Scholar
  3. 3.
    Blanchet, J., Chen, X.: Steady-state simulation of reflected Brownian motion and related stochastic networks (2013). arXiv preprint arXiv:1202.2062
  4. 4.
    Blanchet, J., Dong, J.: Perfect sampling for infinite server and loss systems. Adv. Appl. Probab. 47(3), 761–786 (2014). ForthcomingCrossRefGoogle Scholar
  5. 5.
    Blanchet, J., Sigman, K.: On exact sampling of stochastic perpetuities. J. Appl. Probab. 48(A), 165–182 (2011)CrossRefGoogle Scholar
  6. 6.
    Blanchet, J., Wallwater, A.: Exact sampling for the stationary and time-reversed queues. ACM Trans. Model. Comput. Simul. (TOMACS) 25(4), 26:1–26:27 (2015)CrossRefGoogle Scholar
  7. 7.
    Chen, H., Yao, D.: Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization, vol. 46. Springer, Berlin (2013)Google Scholar
  8. 8.
    Connor, S., Kendall, W.: Perfect simulation for a class of positive recurrent Markov chains. Ann. Appl. Probab. 17(3), 781–808 (2007)CrossRefGoogle Scholar
  9. 9.
    Connor, S., Kendall, W.: Perfect simulation of M/G/c queues. Adv. Appl. Probab. 47(4), 1039–1063 (2015)CrossRefGoogle Scholar
  10. 10.
    Corcoran, J., Tweedie, R.: Perfect sampling of ergodic Harris chains. Ann. Appl. Probab. 11(2), 438–451 (2001)CrossRefGoogle Scholar
  11. 11.
    Ensor, K., Glynn, P.: Simulating the maximum of a random walk. J. Stat. Plan. Inference 85, 127–135 (2000)CrossRefGoogle Scholar
  12. 12.
    Foss, S.: On the approximation of multichannel service systems. Sibirsk. Mat. Zh. 21(6), 132–140 (1980)Google Scholar
  13. 13.
    Foss, S., Chernova, N.: On optimality of the FCFS discipline in multiserver queueing systems and networks. Sib. Math. J. 42(2), 372–385 (2001)CrossRefGoogle Scholar
  14. 14.
    Foss, S., Konstantopoulos, T.: Lyapunov function methods. Lecture Notes. http://www2.math.uu.se/~takis/L/StabLDC06/notes/SS_LYAPUNOV.pdf (2006)
  15. 15.
    Foss, S., Tweedie, R.: Perfect simulation and backward coupling. Stoch. Models 14, 187–203 (1998)CrossRefGoogle Scholar
  16. 16.
    Garmarnik, D., Goldberg, D.: Steady-state GI/GI/n queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23, 2382–2419 (2013)CrossRefGoogle Scholar
  17. 17.
    Hillier, F.S., Lo, F.D.: Tables for multiple-server queueing systems involving Erlang distributions. Tech. Rep. 31, Department of Operations Research, Stanford University (1971)Google Scholar
  18. 18.
    Kelly, F.: Reversibility and Stochastic Networks, vol. 40. Wiley, Chichester (1979)Google Scholar
  19. 19.
    Kendall, W.: Perfect simulation for the area-interaction point process. In: Accardi, L., Heyde, C.C. (eds.) Probability towards 2000, pp. 218–234. Springer, New York (1998)CrossRefGoogle Scholar
  20. 20.
    Kendall, W.: Geometric ergodicity and perfect simulation. Electron. Comm. Probab. 9, 140–151 (2004)CrossRefGoogle Scholar
  21. 21.
    Kendall, W., Møller, J.: Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Probab. 32(3), 844–865 (2000)CrossRefGoogle Scholar
  22. 22.
    Liu, Z., Nain, P., Towsley, D.: Sample path methods in the control of queues. Queueing Syst. 21(1–2), 293–335 (1995)CrossRefGoogle Scholar
  23. 23.
    Propp, J., Wilson, D.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Alg. 9, 223–252 (1996)CrossRefGoogle Scholar
  24. 24.
    Rubinstein, R., Kroese, D.: Simulation and the Monte Carlo method, vol. 707. Wiley, New York (2011)Google Scholar
  25. 25.
    Sigman, K.: Exact simulation of the stationary distribution of the FIFO M/G/c queue. J. Appl. Probab. 48A, 209–216 (2011)CrossRefGoogle Scholar
  26. 26.
    Sigman, K.: Exact sampling of the stationary distribution of the FIFO M/G/c queue: the general case for \(\rho <c\). Queueing Syst. 70, 37–43 (2012)CrossRefGoogle Scholar
  27. 27.
    Wolff, R.: An upper bound for multi-channel queues. J. Appl. Probab. 14, 884–888 (1977)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Management Science and EngineeringStanford UniversityStanfordUSA
  2. 2.Graduate School of BusinessColumbia UniversityNew YorkUSA
  3. 3.Department of IEORColumbia UniversityNew YorkUSA

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