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

TL;DR

This paper analyzes the $G/GI/N$ queue in the Halfin--Whitt regime, deriving both deterministic and stochastic approximations for the queue length process, and extends the diffusion approximation to general service time distributions.

Contribution

It provides the first-order fluid limit and a second-order stochastic approximation for the queue in the Halfin--Whitt regime, including a new characterization involving renewal functions.

Findings

Deterministic fluid limit for the queue length process.

Second-order stochastic approximation involving renewal functions.

Extension of diffusion approximation to general service time distributions.

Abstract

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.