Heavy-Traffic Limits for the G/H2*/n/mQueue

We establish heavy-traffic stochastic-process limits for queue-length, waiting-time and overflow stochastic processes in a class ofG/GI/n/m queueing models withn servers andm extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. 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 byH2*, which are mixtures of an exponential distribution with probabilityp 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, Heavy-traffic limits for queues with many exponential servers,Oper. Res.29 567-588), Puhalskii and Reiman (2000, The multiclassGI/PH/N queue in the Halfin-Whitt regime.Adv. Appl. Probab.32 564-595), and Garnett, Mandelbaum, and Reiman (2002. Designing a call center with impatient customers.Manufacturing Service Oper. Management,4 208-227), we consider a sequence of queueing models indexed by the number of servers,n, and letn tend to infinity along with the traffic intensities ? nso that v n (1 - ? n ) ?for -8 << 8. To treat finite waiting rooms, we letm n v n ? ? for 0 < ? < 8. With the specialH2* 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 establish a limit for theG/M/n/m+ M model, having exponential customer abandonments.