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

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.