The Annals of Applied Probability

Many-server diffusion limits for G/Ph/n+GI queues

J. G. Dai, Shuangchi He, and Tolga Tezcan

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This paper studies many-server limits for multi-server queues that have a phase-type service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded G/Ph/n+GI queues, where the patience times are independent and identically distributed following a general distribution. The next limit theorem is for overloaded G/Ph/n+M queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusion-scaled total-customer-count and server-allocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers n goes to infinity. In the overloaded case, the limit is a multi-dimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for GI/Ph/n queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for M/M/n/+M queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the standard random-time-change theorem.

Article information

Ann. Appl. Probab. Volume 20, Number 5 (2010), 1854-1890.

First available in Project Euclid: 25 August 2010

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Zentralblatt MATH identifier

Primary: 90B20: Traffic problems 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Multi-server queues customer abandonment many-server heavy traffic Halfin–Whitt regime quality and efficiency-driven regime efficiency-driven regime phase-type distribution


Dai, J. G.; He, Shuangchi; Tezcan, Tolga. Many-server diffusion limits for G / Ph / n + GI queues. Ann. Appl. Probab. 20 (2010), no. 5, 1854--1890. doi:10.1214/09-AAP674.

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