Heavy - Traffic Limits for the G / H ∗

We establish heavy-traffic stochastic-process limits for queue-length, waiting-time and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. In order to capture the impact of the service-time distribution beyond its mean within a Markovian framework, we consider a special class of service-time distributions, denoted by H∗ 2 , which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These service-time distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981), Puhalskii and Reiman (2000) and Garnett, Mandelbaum and Reiman (2000), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities ρn so that √ n(1 − ρn) → β for −∞ < β < ∞. To treat finite waiting rooms, we let mn/ √ n → κ for 0 < κ ≤ ∞. With the special H∗ 2 service-time distribution, the limit processes are one-dimensional Markov processes, behaving like diffusion processes with different drift and diffusion functions in two different regions, above and below zero. We also consider the case of exponential customer abandonments for the G/M/n/m model. American Mathematical Society 1991 subject classifications. Primary 60K25, 60F17; Secondary 60K30.