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

Braverman, Anton; Dai, J. G.; Feng, Jiekun

doi:10.1214/15-ssy212

Cited by 13 publications

(25 citation statements)

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“…That is, to ensure that the error bound goes to 0, we require that the number of servers N goes to

\infty

and the service rate μ goes to 0 at the same time , which is supported by our numerical results. This is a main difference from the continuous‐time queueing systems studied in the literature, including Braverman (), where the author develops a state‐dependent diffusion model for Erlang‐C queue. In the continuous‐time queues, just

N, R \to \infty

ensures the convergence of the error bound.…”

Section: Introductionmentioning

confidence: 96%

“…In addition, since the diffusion coefficient a ( x ) is state‐dependent, the density p ( x ) becomes complicated. We apply a new technique developed in Braverman () to establish the gradient bounds. This application is not trivial since our diffusion coefficient has a quadratic form, while the one in Braverman () has a linear form; see details in Sections 1.4 and 3.2.…”

Section: Introductionmentioning

confidence: 99%

“…Stein's method is a well‐known method for establishing error bounds in various fields and applications; see, for example, the survey papers (Chatterjee, ; Ross, ). The proof in our article is mainly based on the framework developed in Braverman and Dai () on applying Stein's method to steady‐state diffusion approximations in

M / P h / N + M

queue; also see a tutorial on applying this framework to Erlang‐A and Erlang‐C queues in Braverman, Dai, and Feng () and the references there for this line of work. Another relevant work is done by Gurvich (), who independently develops a method to prove a steady‐state convergence that is similar to Theorem 1 in Braverman and Dai ().…”

Section: Introductionmentioning

confidence: 99%

“…Among this line of research, the most relevant work is Chapter 3 of doctoral thesis (Braverman, ) (originally appeared as a working paper, Braverman & Dai, ). The author develops a new diffusion model with state‐dependent diffusion coefficients for Erlang‐C queue; this refined diffusion model allows establishing an error bound with a higher order convergence rate.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, this work is a main driver for us to look into the state‐dependent diffusion approximation in the discrete queue. As discussed in Section 1.2, the limit regime we identify is different from that in Braverman (), and we overcome a major challenge in proving the moment bounds that uniquely arises in our discrete setting. Note that to establish the gradient bounds in Section 3.2, we build on a new technique developed in Braverman ().…”

Section: Introductionmentioning

confidence: 99%

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Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

Feng

Shi

2018

Naval Research Logistics

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In this article, we analyze a discrete‐time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady‐state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state‐dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions—from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.

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“…That is, to ensure that the error bound goes to 0, we require that the number of servers N goes to

\infty

N, R \to \infty

ensures the convergence of the error bound.…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

M / P h / N + M

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

Feng

Shi

2018

Naval Research Logistics

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Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

Ebner

2019

Ann Inst Stat Math

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By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein's method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavours, we focus on explicit representations given through a formula for the density-or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known.To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.MSC 2010 subject classifications. Primary 62E10 Secondary 60E10, 62G10Over the last decades, Stein's method for distributional approximation has become a viable tool for proving limit theorems and establishing convergence rates. At it's heart lies the well-known Stein characterization which states that a real-valued random variable Z has a standard normal distribution if, and only if,holds for all functions f of a sufficiently large class of test functions. To exploit this characterization for testing the hypothesisof normality, where P X is the distribution of a real-valued random variable X, against general alternatives, Betsch and Ebner (2019b) used that (1.1) can be untied from the class of test functions with the help of the so-called zero-bias transformation introduced by Goldstein and Reinert (1997). To be specific, a real-valued random variable X * is said to have the X-zero-bias distri-bution ifholds for any of the respective test functions f . If EX = 0 and Var(X) = 1, the X-zero-bias distribution exists and is unique, and it has distribution function(1.3)By (1.1), the standard Gaussian distribution is the unique fixed point of the transformation P X → P X * . Thus, the distribution of X is standard normal if, and only if,where F X denotes the distribution function of X. In the spirit of characterization-based goodnessof-fit tests, an idea introduced by Linnik (1962), this fixed point property directly admits a new class of testing procedures as follows. Letting T X n be an empirical version of T X and F n the empirical distribution function, both based on the standardized sample, Betsch and Ebner (2019b) proposed a test for (1.2) based on the statisticwhere w is an appropriate weight function, which, in view of (1.4), rejects the normality hypothesis for large values of G n . As these tests have several desirable properties such as consistency against general alternatives, and since they show a very promising performance in simulations, we devote this work to the question to what extent the fixed point property and the class of goodness-of-fit procedures may be generalized to othe...

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

Gaunt

Walton

2020

Statistics & Probability Letters

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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]...

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

Cited by 13 publications

References 66 publications

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

Fixed point characterizations of continuous univariate probability distributions and their applications

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

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