Spectral Gap of the Erlang A Model in the Halfin-Whitt Regime

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 HalfinWhitt 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.1. 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]). The Halfin-Whitt regime in fact 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. For such systems with abandonments, we are interested in the spectral gap, which is inversely related to the relaxation time or the speed at which a system reaches stationarity. A large relaxation time in general indicates that replacing time-dependent characteristics by their stationary counterparts might

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“…Thus, it comes as no surprise that the Erlang-A model, which is a Markovian and multiserver queueing model that incorporates customer abandonments, is an important modeling tool in a multitude of application settings. Some of the more prominent applications include telecommunications and call/contact centers, healthcare, urban mobility and transportation, and more recently cloud computing; see, for example, [28], [29], [45], [37], [3], [5], [13], [48], [44], [2], [41], [20], [42], [12], and [4]. A detailed overview of the model can be found in [26].…”