Annals of Applied Probability

The G/GI/N queue in the Halfin–Whitt regime

Josh Reed

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In this paper, we study the G/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.

Article information

Ann. Appl. Probab., Volume 19, Number 6 (2009), 2211-2269.

First available in Project Euclid: 25 November 2009

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory [See also 68M20, 90B22] 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 60G15: Gaussian processes 60G44: Martingales with continuous parameter 60K15: Markov renewal processes, semi-Markov processes

Queueing theory diffusion approximation Gaussian process martingale weak convergence


Reed, Josh. The G / GI / N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19 (2009), no. 6, 2211--2269. doi:10.1214/09-AAP609.

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