## 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/√λ, where the constant C is independent of the arrival rate λ 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 mth 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.

Original language | English (US) |
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Pages (from-to) | 550-581 |

Number of pages | 32 |

Journal | Annals of Applied Probability |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2017 |

## Keywords

- Convergence rate
- Diffusion approximation
- Many servers
- State space collapse
- Steady-state
- Stein's method

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty