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Perfect simulation of M/G/c queues

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

Published online by Cambridge University Press:  21 March 2016

Stephen B. Connor  and
Wilfrid S. Kendall
Stephen B. Connor*
Affiliation:
University of York
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Mathematics, University of York, York YO10 5DD, UK. Email address: stephen.connor@york.ac.uk
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

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In this paper we describe a perfect simulation algorithm for the stable M/G/c queue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/c queue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/c queue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/c queues started respectively from the empty state and the state of the M/G/c queue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.

Keywords

Coalescencedominated coupling-from-the-pastdynamic reversibilityfirst-come-first-served disciplinefirst-in-first-out disciplineKiefer-Wolfowitz workload vectorpathwise dominationperfect simulationprocessor sharing disciplineM/G/c queuerandom assignment disciplinesandwichingstable queuestochastic orderingsuper-stable queue

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60K25: Queueing theory 60J05: Discrete-time Markov processes on general state spaces 68U20: Simulation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 1039 - 1063
Copyright
Copyright © Applied Probability Trust 2015 

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