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February 2017 Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems
Anton Braverman, J. G. Dai
Ann. Appl. Probab. 27(1): 550-581 (February 2017). DOI: 10.1214/16-AAP1211


We consider $M/\mathit{Ph}/n+M$ queueing systems in steady state. We prove that the Wasserstein distance between the stationary distribution of the normalized system size process and that of a piecewise Ornstein–Uhlenbeck (OU) process is bounded by $C/\sqrt{\lambda}$, where the constant $C$ is independent of the arrival rate $\lambda$ and the number of servers $n$ as long as they are in the Halfin-Whitt parameter regime. For each integer $m>0$, we also establish a similar bound for the difference of the $m$th steady-state moments. For the proofs, we develop a modular framework that is based on Stein’s method. The framework has three components: Poisson equation, generator coupling, and state space collapse. The framework, with further refinement, is likely applicable to steady-state diffusion approximations for other stochastic systems.


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Anton Braverman. J. G. Dai. "Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems." Ann. Appl. Probab. 27 (1) 550 - 581, February 2017.


Received: 1 March 2015; Revised: 1 November 2015; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1362.60077
MathSciNet: MR3619795
Digital Object Identifier: 10.1214/16-AAP1211

Primary: 60K25
Secondary: 60F99 , 60J60 , 90B20

Keywords: convergence rate , diffusion approximation , many servers , state space collapse , steady-state , Stein’s method

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 1 • February 2017
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