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

TL;DR

This paper derives diffusion limit theorems for large multi-server queues with phase-type services and customer abandonment, extending existing models to more general and overloaded scenarios with innovative proof techniques.

Contribution

It provides new diffusion limit theorems for $G/Ph/n+GI$ queues, including critically loaded and overloaded cases, generalizing prior results and introducing novel proof methods.

Findings

Diffusion limits for critically loaded queues generalize Puhalskii and Reiman (2000).

Overloaded queue limits refine fluid models and extend Whitt (2004).

Innovative proof techniques involve system perturbation and mapping methods.

Abstract

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.