A diffusion approximation for the G/Gl/n/m queue
成果类型：
Article
署名作者：
Whitt, W
署名单位：
Columbia University
刊物名称：
OPERATIONS RESEARCH
ISSN/ISSBN：
0030-364X
DOI：
10.1287/opre.1040.0136
发表日期：
2004
页码：
922-941
关键词：
摘要：
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/infinity special case, Halfin and Whitt (1981) showed that scaled versions of the queue-length process converge to a diffusion process when the traffic intensity rho(n) approaches 1 with (1 - rho(n))rootn --> beta for 0 < beta < infinity. 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 I - p. Finite waiting rooms are treated by incorporating the additional limit m(n)/rootn --> kappa for 0 < kappa less than or equal to infinity. 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/infinity model with phase-type service-time distributions established by Puhalskii and Reiman (2000) imply that our approximating process is not asymptotically correct for non-exponential 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.