Research Details

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 well that of the Markov chain. 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 is proved to guarantee that the gap between steady-state moments of the diffusion and those of the properly centered and scaled CTMCs, shrinks at a rate of v n. The uniform Lyapunov requirement is satisfied, in particular, if the scaled and centered sequence converges to a diffusion limit for which a Lyapunov condition is satisfied. Our proofs build on gradient estimates for the solutions of the Poisson equations associated with the (sequence of) diffusion models together with 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.

Type

Article

Author(s)

Itai Gurvich

Date Published

2014

Citations

Gurvich, Itai. 2014. Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. Annals of Applied Probability.(6): 2527-2559.