Spectral gap of the Erlang A model in the Halfin-Whitt regime

Johan S H Van Leeuwaarden,Charles Knessl

doi:10.1214/10-ssy012

Johan S H Van Leeuwaarden, Charles Knessl

Open Access

https://doi.org/10.1214/10-ssy012

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Abstract

We consider a hybrid diffusion process that is a combination of two Ornstein-Uhlenbeck processes with different restraining forces. This process serves as the heavy-traffic approximation to the Markovian many-server queue with abandonments in the critical Halfin-Whitt regime. We obtain an expression for the Laplace transform of the time-dependent probability distribution, from which the spectral gap is explicitly characterized. The spectral gap gives the exponential rate of convergence to equilibrium. We further give various asymptotic results for the spectral gap, in the limits of small and large abandonment effects. It turns out that convergence to equilibrium becomes extremely slow for overloaded systems with small abandonment effects.

Highlights

Within the fields of stochastic processes and queueing theory, the Halfin-Whitt regime refers to a mathematical way of establishing economies-of-scale in many-server queueing systems like call centers
The Halfin-Whitt regime prescribes a scaling under which the many-server systems converge to limiting processes, which are for most systems diffusion processes
This paper deals with many-server systems in the Halfin-Whitt regime with the additional feature that customers are impatient, and may abandon the system without being served

Summary

Introduction

Within the fields of stochastic processes and queueing theory, the Halfin-Whitt regime refers to a mathematical way of establishing economies-of-scale in many-server queueing systems like call centers (see [13]). We further give various asymptotic results for the spectral gap, in the limits of small and large abandonment effects.

Results

Conclusion