Stein’s method for the single server queue in heavy traffic

Abstract

Following recent developments in the application of Stein's method in queueing theory, this paper is intended to be a short treatment showing how Stein's method can be developed and applied to the single server queue in heavy traffic. Here we provide two approaches to this approximation: one based on equilibrium couplings and another involving comparison of generators.an overview can be found in the survey Ross [27]. We summarise and apply two such approaches to the stationary single server queue.Stein's method has found applicability in a number of areas such as random graphs [2], branching processes [23] and statistical mechanics [14]; see Ross [27] for a recent review of applications and methods. However, only recently has Stein's method begun to be applied to queueing theory. Specifically, following the work of Gurvich [18], Braveman and Dai in a series of papers and another together with Feng ascertained and developed the application of Stein's method in queueing using a Basic Adjoint Relation (BAR) approach [5,4,6,7]. These works principally provide approximations between Erlang queueing models and their limiting stationary distributions in the Halfin-Whitt asymptotic. As noted above, another limiting regime is Heavy Traffic. Braverman, Dai and Myazawa [8] apply their BAR approach to prove weak convergence of stationary distributions in Heavy Traffic. Recently, Besançon, Decreusefond, and Moyal [3] have used Stein's method to obtain explicit bounds for the diffusion approximations for the number of customers in the M/M/1 and M/M/∞ queues. Their results, which are obtained using the functional Stein's method introduced for the Brownian approximation of Poisson processes [12], differ from the aforementioned results as they are given at the process level.From the famous Pollaczek-Khinchine formula for moment-generating functions or via ladder-height arguments, it can be shown that the stationary waiting time distribution of the M/G/1 and G/G/1 queues can be expressed as a geometric convolution, that is the sum of a geometrically distributed number of IID random variables.We review a number of works that consider exponential approximations or geometric convolutions. The work of Brown [9] finds approximations of geometric convolutions to the exponential distribution using renewal theory techniques, rather than Stein's method. More recent work of Brown [10] improves upon these bounds under certain hazard rate assumptions. Recent works of Peköz and Röllin [23] and Peköz, Röllin and Ross [24] apply Stein's method to the exponential and geometric approximations respectively. Theorem 3.1, below, is analogous to Theorem 3.1 of [23]. Contemporaneously with the work of Braverman and Dai, Daly [13] applies Stein's method to quantify the approximation between geometric convolutions and non-negative integer valued random variables.Consider now the M/G/1 queue with inter-arrival times following the Exp(λ) distribution and a general service time distribution S. We let W denote its stationary waiting time and define ρ = λE[S]...

Transform Methods for Heavy-Traffic Analysis

“…Universal approximations for queues with abandonment were obtained using Stein's method by Huang and Gurvich (2018). More recently, a single-server queue in heavy traffic was studied using Stein's method by Gaunt and Walton (2020). Gurvich et al (2013) studies Erlang-A system and obtains universal approximations using excursion-based analysis as opposed to using Stein's method.…”

The Prelimit Generator Comparison Approach of Stein’s Method

“…It is also well known that as ρ → 1, the customer count can be approximated by an exponential random variable. A recent application of Stein's method in Gaunt and Walton (2020) establishes convergence rates of the waiting time distribution in the M=G=1 system (which is more general than the M=M=1 system) to the exponential distribution.…”

The join-the-shortest-queue system in the Halfin-Whitt regime: rates of convergence to the diffusion limit

“…Popularized in the area of queueing systems by Gurvich (2014), Ying (2017), Braverman and Dai (2017), Gast (2017), the generator comparison approach of Stein's method, attributed to Barbour (1988Barbour ( , 1990 and Götze (1991), is used to study convergence rates of steadystate Markov chain distributions to their diffusion, fluid, or mean-field limits. For a few recent applications of the generator comparison approach in queueing, we refer the reader to Gaunt and Walton (2020), Hurtado-Lange and Maguluri (2021), Lu (2021), Liu et al (2022); this list is by no means comprehensive. In this paper, we restrict our attention to the case when the limit is the stationary distribution of a diffusion process, referring the reader to Ying (2017) for a treatment of fluid and mean-field limits.…”