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

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 ρ_n approaches 1 with (1 – ρ_n)√n → β for 0 < β < ∞. 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 → κ for 0 < κ ≤ ∞. 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.