Heavy-Traffic Limits for the G / H 2 */ n / m Queue

Source abstract

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. 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, Heavy-traffic limits for queues with many exponential servers, Oper. Res. 29 567–588), Puhalskii and Reiman (2000, The multiclass GI/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 let n tend to infinity along with the traffic intensities Ρ n so that √n (1 − Ρ n ) → β for −∞ < β < ∞. To treat finite waiting rooms, we let m n √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 establish a limit for the G/M/n/m+M model, having exponential customer abandonments.