Diffusion models and steady-State approximations for exponentially ergodic Markovian queues

Itai Gurvich*

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

31 Scopus citations


Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove, approximates that of the Markov chain with notable precision. Strong approximations provide such "limitless" approximations for process dynamics. Our focus here is on steady-state distributions, and the diffusion model that we propose is tractable relative to strong approximations. Within an asymptotic framework, in which a scale parameter n is taken large, a uniform (in the scale parameter) Lyapunov condition imposed on the sequence of diffusion models guarantees that the gap between the steady-state moments of the diffusion and those of the properly centered and scaled CTMCs shrinks at a rate of √n. Our proofs build on gradient estimates for solutions of the Poisson equations associated with the (sequence of) diffusion models and on elementary martingale arguments. As a by-product of our analysis, we explore connections between Lyapunov functions for the fluid model, the diffusion model and the CTMC.

Original languageEnglish (US)
Pages (from-to)2527-2559
Number of pages33
JournalAnnals of Applied Probability
Issue number6
StatePublished - Dec 1 2014


  • Halfin-Whitt regime
  • Heavy-traffic
  • Many servers
  • Markovian queues
  • Steady state approximations
  • Steady-state
  • Strong approximations for queues

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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