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

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.

Published in at http://dx.doi.org/10.1214/09-AAP674 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Martingale proofs of many-server heavy-traffic limits for Markovian queues
• The $G/GI/N$ queue in the Halfin--Whitt regime
• Proofs of the martingale FCLT
• Fluid limits of many-server queues with reneging
• On many-server queues in heavy traffic

Cited by in corpus (11)

• Diffusion models and steady-state approximations for exponentially ergodic Markovian queues
• Steady-state $GI/GI/n$ queue in the Halfin-Whitt regime
• Fluid Models of Many-server Queues with Abandonment
• Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues
• Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions
• Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems
• On the steady-state probability of delay and large negative deviations for the $GI/GI/n$ queue in the Halfin-Whitt regime
• An approximation to steady-state of M/Ph/n+M queue
• A Non-convex Optimization Approach of Searching Algebraic Degree Phase-type Representations for General Phase-type Distributions
• A one-dimensional diffusion model for overloaded queues with customer abandonment
• Exponential ergodicity and steady-state approximations for a class of Markov processes under fast regime switching