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

Transient analysis of the Erlang A model

“…In this limit we can approximate the contour integrals , , and by simpler special functions, namely parabolic cylinder functions. We discuss this limit in detail in van Leeuwaarden and Knessl (2011) for the M  /  M  /  m model with , and in Leeuwaarden and Knessl (2012) for the model with . We can obtain then where P will satisfy a parabolic PDE, which we explicitly solved in van Leeuwaarden and Knessl (2011), Leeuwaarden and Knessl (2012).…”

“…We discuss this limit in detail in van Leeuwaarden and Knessl (2011) for the M  /  M  /  m model with , and in Leeuwaarden and Knessl (2012) for the model with . We can obtain then where P will satisfy a parabolic PDE, which we explicitly solved in van Leeuwaarden and Knessl (2011), Leeuwaarden and Knessl (2012). An alternate approach is to evaluate Theorems 1 and 2, or Corollary 3 in the limit in (2.104), and thus identify P ( x ,  t ) directly.…”

“…Both the stationary behavior (Garnett et al. 2002) and the time-dependent behavior (Leeuwaarden and Knessl 2012) of this process are well understood. From our general result for the Laplace transform of we show how the results obtained in Leeuwaarden and Knessl (2012) for the above diffusion processes can be recovered.…”

“…2002) and the time-dependent behavior (Leeuwaarden and Knessl 2012) of this process are well understood. From our general result for the Laplace transform of we show how the results obtained in Leeuwaarden and Knessl (2012) for the above diffusion processes can be recovered. See the survey paper Ward (2012) for a comprehensive overview of diffusion approximations for many-server systems with abandonments.…”

Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

“…Now consider an asymptotic regime where the number of servers grows large, and additionally assume that N − λ(N) √ N → β as N → ∞ for some positive coefficient β > 0, i.e., the load per server approaches unity as 1 − β/ √ N. In terms of the aggregate traffic load and total service capacity, this scaling corresponds to the socalled Halfin-Whitt heavy-traffic regime which was introduced in the seminal paper [12] and has been extensively studied since. The set-up in [12], as well as the numerous model extensions in the literature (see [8,9,10,12,23,24,25], and the references therein), primarily considered a single centralized queue and server pool (M/M/N), rather than a scenario with parallel queues. Eschenfeldt and Gamarnik [7] initiated the study of the scaling behavior for parallel-server systems in the Halfin-Whitt heavy-traffic regime.…”