The G/GI/N Queue in the Halfin-Whitt Regime I: Infinite Server Queue System Equations

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 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 [6] in the case of exponentially distributed service times. AMS 2000 Subject Classification60G15,60G44,60K25 Keyword and Phrases queueing theory, diffusion approximation, Gaussian process, martingale, weak convergence