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
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
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