Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems

Abstract

We consider $M/Ph/n+M$ 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.

References in corpus (3)

• Validity of heavy traffic steady-state approximations in generalized Jackson Networks
• Many-server diffusion limits for $G/Ph/n+GI$ queues
• A short survey of Stein's method

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• Diffusive limits of Lipschitz functionals of Poisson measures
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• Heavy-tailed queues in the Halfin-Whitt regime
• On the Rate of Convergence of Mean-Field Models: Stein's Method Meets the Perturbation Theory
• Asymptotic dimensioning of stochastic service systems