This Article Full Text (PDF) References Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Whitt, W. Search for Related Content

A Diffusion Approximation for the G/GI/n/m Queue

Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027-6699
ward.whitt{at}columbia.edu

We develop a diffusion approximation for the queue-length stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra waiting spaces). We use the steady-state distribution of that diffusion process to obtain approximations for steady-state performance measures of the queueing model, focusing especially upon the steady-state delay probability. The approximations are based on heavy-traffic limits in which n tends to infinity as the traffic intensity increases. Thus, the approximations are intended for large n.

For the GI/M/n/ special case, Halfin and Whitt (1981) showed that scaled versions of the queue-length process converge to a diffusion process when the traffic intensity _n approaches 1 with (1 – _n)n ß for 0 < ß < . A companion paper, Whitt (2005), extends that limit to a special class of G/GI/n/m_n models in which the number of waiting places depends on n and the service-time distribution is a mixture of an exponential distribution with probability p and a unit point mass at 0 with probability 1 – p. Finite waiting rooms are treated by incorporating the additional limit m_n/n for 0 < . The approximation for the more general G/GI/n/m model developed here is consistent with those heavy-traffic limits. Heavy-traffic limits for the GI/PH/n/ model with phase-type service-time distributions established by Puhalskii and Reiman (2000) imply that our approximating process is not asymptotically correct for nonexponential phase-type service-time distributions, but nevertheless, the heuristic diffusion approximation developed here yields useful approximations for key performance measures such as the steady-state delay probability. The accuracy is confirmed by making comparisons with exact numerical results and simulations.

Subject classifications: queues; approximations; multiserver queues; queues; multichannel; diffusion approximation.
History: Received July 2002; revision received March 2003; accepted September 2003.

This article has been cited by other articles:

				A. Mandelbaum and P. Momcilovic Queues with Many Servers: The Virtual Waiting-Time Process in the QED Regime Mathematics of Operations Research, August 1, 2008; 33(3): 561 - 586. [Abstract] [PDF]

				T. Tezcan Optimal Control of Distributed Parallel Server Systems Under the Halfin and Whitt Regime Mathematics of Operations Research, February 1, 2008; 33(1): 51 - 90. [Abstract] [PDF]

				J. M. Milner and T. L. Olsen Service-Level Agreements in Call Centers: Perils and Prescriptions Management Science, February 1, 2008; 54(2): 238 - 252. [Abstract] [PDF]

				N. Gans and Y.-P. Zhou Call-Routing Schemes for Call-Center Outsourcing MSOM, January 1, 2007; 9(1): 33 - 50. [Abstract] [PDF]

				W. Whitt Heavy-Traffic Limits for the G/H2*/n/mQueue Mathematics of Operations Research, February 1, 2005; 30(1): 1 - 27. [Abstract] [PDF]

				W. Whitt Engineering Solution of a Basic Call-Center Model Management Science, February 1, 2005; 51(2): 221 - 235. [Abstract] [PDF]