The $G/GI/N$ queue in the Halfin--Whitt regime

Details The $G/GI/N$ queue in the Halfin--Whitt regime The $G/GI/N$ queue in the Halfin--Whitt regime

2009-12-15

arxiv.org/abs/0912.2837v1

Annals of Applied Probability 2009, Vol. 19, No. 6, 2211-2269

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP609 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP609

In this paper, we study the $G/\mathit{GI}/N$ queue in the Halfin--Whitt regime. Our first result is to obtain a deterministic fluid limit for the properly centered and scaled number of customers in the system which may be used to provide a first-order approximation to the queue length process. Our second result is to obtain a second-order stochastic approximation to the number of customers in the system in the Halfin--Whitt regime. This is accomplished by first centering the queue length process by its deterministic fluid limit and then normalizing by an appropriate factor. We then proceed to obtain an alternative but equivalent characterization of our limiting approximation which involves the renewal function associated with the service time distribution. This alternative characterization reduces to the diffusion process obtained by Halfin and Whitt [Oper. Res. 29 (1981) 567--588] in the case of exponentially distributed service times.

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