Skip to main content
×
×
Home

Perfect sampling for infinite server and loss systems

  • Jose Blanchet (a1) and Jing Dong (a1)
Abstract

We present the first class of perfect sampling (also known as exact simulation) algorithms for the steady-state distribution of non-Markovian loss systems. We use a variation of dominated coupling from the past. We first simulate a stationary infinite server system backwards in time and analyze the running time in heavy traffic. In particular, we are able to simulate stationary renewal marked point processes in unbounded regions. We then use the infinite server system as an upper bound process to simulate the loss system. The running time analysis of our perfect sampling algorithm for loss systems is performed in the quality-driven (QD) and the quality-and-efficiency-driven regimes. In both cases, we show that our algorithm achieves subexponential complexity as both the number of servers and the arrival rate increase. Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of complexity.

Copyright
Corresponding author
Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA. Email address: jose.blanchet@columbia.edu
∗∗ Current address: Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208, USA. Email address: jing.dong@northwestern.edu
References
Hide All
[1] Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes (IMS Lecture Notes Monogr. Ser. 12), IMS, Hayward, CA.
[2] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Spinger, New York.
[3] Berthelsen, K. K. and M⊘ller, J. (2002). A primer on perfect simulation for spatial point processes. Bull. Braz. Math. Soc. (N.S.) 33, 351367.
[4] Blanchet, J. and Dong, J. (2012). Sampling point processes on stable unbounded regions and exact simulation of queues. In Proc. 2012 Winter Simulation Conference, IEEE, New York, pp. 112.
[5] Blanchet, J. and Lam, H. (2014). Rare-event simulation for many-sever queues. Math. Operat. Res. 39, 11421178.
[6] Blanchet, J. and Sigman, K. (2011). On exact sampling of stochastic perpetuities. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A, Applied Probability Trust, Sheffield, pp. 165182.
[7] Blanchet, J., Chen, X. and Lam, H. (2014). Two-parameter sample path large deviations for infinite-server queues. Stoch. Systems 4, 206249.
[8] Brown, L. et al. (2005). Statistical analysis of a telephone call center: a queueing-science perspective. J. Amer. Statist. Assoc. 100, 3650.
[9] Bušić, A., Gaujal, B. and Perronnin, F. (2012). Perfect sampling of networks with finite and infinite capacity queues. In Analytical and Stochastic Modeling Techniques and Applications (Lecture Notes Comput. Sci. 7314), Springer, Berlin, pp. 136-149.
[10] Connor, S. B. and Kendall, W. S. (2007). Perfect simulation for a class of positive recurrent Markov chains. Ann. Appl. Prob. 17, 781808. %ED: is this double page range correct—check?
[11] Corcoran, J. N. and Tweedie, R. L. (2001). Perfect sampling of ergodic Harris chains. Ann. Appl. Prob. 11, 438451.
[12] Dong, J. (2014). Studies in stochastoc networks: efficient Monte-Carlo methods, modeling and asymptotic analysis. Doctoral Thesis. Columbia University.
[13] Ensor, K. B. and Glynn, P. W. (2000). Simulating the maximum of a random walk. J. Statist. Planning Infer. 85, 127135.
[14] Ferrari, P. A., Fernández, R. and Garcia, N. L. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 6388.
[15] Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Commun. Statist. Stoch. Models 14, 187203.
[16] Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319378.
[17] Kendall, W. S. (1995). Perfect simulation for area-interaction point processes. In Probability Towards 2000, Springer, New York, pp. 218234.
[18] Kendall, W. S. (2004). Geometric ergodicity and perfect simulation. Electron. Commun. Prob. 9, 140151.
[19] Kendall, W. S. and M⊘ller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844865.
[20] Murdoch, D. J. and Takahara, G. (2006). Perfect sampling for queues and network models. ACM Trans. Model. Comput. Simul. 16, 7692.
[21] Pang, G. and Whitt, W. (2010). Two-parameter heavy-traffic limits for infinite-server queues. Queueing Systems 65, 325364.
[22] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223252.
[23] Sigman, K. (2011). Exact simulation of the stationary distribution of the FIFO M/G/c queue. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A), Applied Probability Trust, Sheffield, pp. 209213.
[24] Sigman, K. (2012). Exact simulation of the stationary distribution of the FIFO M/G/c queue: The general case of ρ < c . Queueing Systems 70, 3743.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed