Perfect simulation of M/G/c queues

TY - JOUR

T1 - Perfect simulation of M/G/c queues

AU - Connor, Stephen

AU - Kendall, W.S.

PY - 2016/3/21

Y1 - 2016/3/21

N2 - 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.

AB - 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.

KW - 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 do

U2 - 10.1239/aap/1449859799

DO - 10.1239/aap/1449859799

M3 - Article

SN - 0001-8678

VL - 47

SP - 1039

EP - 1063

JO - Advances in Applied Probability

JF - Advances in Applied Probability

IS - 4

ER -