Perfect simulation of M/G/c queues. (December 2015)

• Record Type: Journal Article
• Authors: Connor, Stephen B.; Kendall, Wilfrid S.
• Is Part Of: Advances in applied probability. Volume 47:Number 4(2015)
• Page Start: 1039
• Page End: 1063
• Publication Date: 2015-12
• Subjects: Coalescence, -- dominated coupling-from-the-past, -- dynamic reversibility, -- first-come-first-served discipline, -- first-in-first-out discipline, -- Kiefer-Wolfowitz workload vector, -- pathwise domination, -- perfect simulation, -- processor sharing discipline, -- M/G/c queue, -- random assignment discipline, -- sandwiching, -- stable queue, -- stochastic ordering, -- super-stable queue; 65C05, -- 60K25, -- 60J05, -- 68U20; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1239/aap/1449859799
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract : 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.