Excursion-Based Universal Approximations for the Erlang-A Queue in Steady-State

Abstract

We revisit many-server approximations for the well-studied Erlang-A queue. This is a system with a single pool of i.i.d. servers that serve one class of impatient i.i.d. customers. Arrivals follow a Poisson process and service times are exponentially distributed as are the customers' patience times. We propose a diffusion approximation that applies simultaneously to all existing many-server heavy-traffic regimes: quality and efficiency driven, efficiency driven, quality driven, and nondegenerate slowdown. We prove that the approximation provides accurate estimates for a broad family of steady-state metrics. Our approach is "metric-free" in that we do not use the specific formulas for the steady-state distribution of the Erlang-A queue. Rather, we study excursions of the underlying birth-and-death process and couple these to properly defined excursions of the corresponding diffusion process. Regenerative process and martingale arguments, together with derivative bounds for solutions to certain ordinary differential equations, allow us to control the accuracy of the approximation. We demonstrate the appeal of universal approximation by studying two staffing optimization problems of practical interest.

Heavy Traffic Approximation for the Stationary Distribution of a Generalized Jackson Network: The BAR Approach

“…In the many-server setting, Tezcan [38] considered a parallel-server system with multiple server pools and no customer abandonment, Gamarnik and Stolyar [16] examined a multiclass, many-server queue with abandonment, where customer service and patience times are exponentially distributed with means varying between different customer classes, and Dai et al [12] considered a many-server queue with abandonment, where service times follow a phase-type distribution. In recent years, several papers [6,7,18,20,24] have gone beyond limit theorems, and establish rates of convergence to the approximating distribution. The framework underlying those papers (except for [20]) is known as Stein's method [37,10,34].…”

“…In recent years, several papers [6,7,18,20,24] have gone beyond limit theorems, and establish rates of convergence to the approximating distribution. The framework underlying those papers (except for [20]) is known as Stein's method [37,10,34].…”

Stein’s method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C models

“…The last regime is the nondegenerate-slowdown (NDS) regime, which was studied in [1,62]. Universal approximations were previously studied in [31,59]. Third, as part of the universality of Theorem 1, we see that…”

Heavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approach

“…In the many-server setting, Tezcan [36] considered a parallel-server system with multiple server pools and no customer abandonment, Gamarnik and Stolyar [16] examined a multiclass, many-server queue with abandonment, where customer service and patience times are exponentially distributed with means varying between different customer classes, and Dai et al [12] considered a many-server queue with abandonment, where service times follow a phase-type distribution. In recent years, several papers [5,6,18,20,23] have gone beyond limit theorems, and establish rates of convergence to the approximating distribution. The framework underlying those papers (except for [20]) is known as Stein's method [35,9,32].…”

“…In recent years, several papers [5,6,18,20,23] have gone beyond limit theorems, and establish rates of convergence to the approximating distribution. The framework underlying those papers (except for [20]) is known as Stein's method [35,9,32].…”