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December 2014 Diffusion models and steady-state approximations for exponentially ergodic Markovian queues
Itai Gurvich
Ann. Appl. Probab. 24(6): 2527-2559 (December 2014). DOI: 10.1214/13-AAP984

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 $\sqrt{n}$.

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.

Citation

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Itai Gurvich. "Diffusion models and steady-state approximations for exponentially ergodic Markovian queues." Ann. Appl. Probab. 24 (6) 2527 - 2559, December 2014. https://doi.org/10.1214/13-AAP984

Information

Published: December 2014
First available in Project Euclid: 26 August 2014

zbMATH: 1333.60194
MathSciNet: MR3262510
Digital Object Identifier: 10.1214/13-AAP984

Subjects:
Primary: 49L20 , 60F17 , 60K25 , 90B20 , 90B36

Keywords: Halfin–Whitt regime , heavy-traffic , many servers , Markovian queues , steady state approximations , steady-state , strong approximations for queues

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 6 • December 2014
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