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

Anton Braverman, J. G. Dai

Research output: Contribution to journalArticlepeer-review

49 Scopus citations


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 languageEnglish (US)
Pages (from-to)550-581
Number of pages32
JournalAnnals of Applied Probability
Issue number1
StatePublished - Feb 2017


  • 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


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