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Exact simulation of the stationary distribution of the FIFO M/G/c queue

Part of: Special processes Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Karl Sigman
Karl Sigman*
Affiliation:
Columbia University, Department of Industrial Engineering and Operations Research, Columbia University, MC: 4704, New York, NY 10027, USA. Email address: karl.sigman@columbia.edu
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Abstract

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We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1. We assume that the service time distribution G(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distribution Ge(x)=μ∫0x P(S>y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

Keywords

Exact simulationcoupling from the pastprocessor sharingqueueing theory

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60K25: Queueing theory 60J05: Discrete-time Markov processes on general state spaces 68U20: Simulation
Type
Part 4. Simulation
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 209 - 213
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S., (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
[2] Asmussen, S. and Glynn, P. W., (2007). Stochastic Simulation. Springer, New York.Google Scholar
[3] Chen, H. and Yao, D. D., (2001). Fundamentals of Queueing Networks. Springer, New York.Google Scholar
[4] Kendall, W., (2004). Geometric ergodicity and perfect simulation. Electron. Commun. Prob. 9, 140–151.Google Scholar
[5] Loynes, R. M., (1962). The stability of a queue with non-independent interarrival and service times. Math. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[6] Propp, J. G. and Wilson, D. B., (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
[7] Ross, S. M., (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
[8] Wolff, R. W., (1987). Upper bounds on work in system for multichannel queues. J. Appl. Prob. 24, 547–551.CrossRefGoogle Scholar