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

Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
ww2040{at}columbia.edu

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 H₂^*, 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 P_n so that n (1 – P_n) ß for – < ß < . To treat finite waiting rooms, we let m_nn for 0 < < . With the special H₂^* 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.

Key Words: queues; multiserver queues; stochastic-process limits; heavy-traffic; Halfin-Whitt regime; diffusion approximations; abandonments; reneging; customer impatience
History: Received: July 17, 2002; revision received: June 30, 2003;revision received: December 22, 2003;revision received: May 26, 2004;

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