The Annals of Applied Probability

Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems

Anton Braverman and J. G. Dai

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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.

Article information

Ann. Appl. Probab., Volume 27, Number 1 (2017), 550-581.

Received: March 2015
Revised: November 2015
First available in Project Euclid: 6 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90B20: Traffic problems 60F99: None of the above, but in this section 60J60: Diffusion processes [See also 58J65]

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


Braverman, Anton; Dai, J. G. Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems. Ann. Appl. Probab. 27 (2017), no. 1, 550--581. doi:10.1214/16-AAP1211.

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