Perfect simulation of M/G/c queues

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JournalAdvances in Applied Probability
DateAccepted/In press - 1 Dec 2015
DatePublished (current) - 21 Mar 2016
Issue number4
Volume47
Number of pages24
Pages (from-to)1039-1063
Original languageEnglish

Abstract

In this paper we describe a perfect simulation algorithm for the stable M/G/c queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how to build a dominated CFTP algorithm for perfect simulation of the super-stable M/G/c queue operating under First Come First Served discipline, with dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when when service durations are coupled in order of initiation of service), and exploiting 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 generalize 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 naive 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 summarize 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.

    Research areas

  • coalescence; coupling; coupling from the past (CFTP); dominated coupling from the past; dynamic reversibility; exact simulation; First Come First Served discipline [FCFS]; First In First Out discipline [FIFO]; KieferWolfowitz workload vector; path-wise domination; perfect simulation; Processor Sharing discipline [PS]; M/G/c queue; Random Assignment discipline [RA]; sandwiching; stable queue; stochastic ordering; super-stable queue; time reversal

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