## Abstract

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 language | English (US) |
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Pages (from-to) | 2527-2559 |

Number of pages | 33 |

Journal | Annals of Applied Probability |

Volume | 24 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2014 |

## Keywords

- 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