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

TL;DR

This paper develops diffusion-based steady-state approximations for Markovian queues with many servers, demonstrating their accuracy and providing a theoretical framework for understanding the convergence of steady-state distributions.

Contribution

It introduces a tractable diffusion model for steady-state approximation of Markov chains, with uniform Lyapunov conditions ensuring convergence rates as the system scale grows.

Findings

Steady-state moments of diffusion and CTMCs converge at rate √n.

Diffusion models provide precise steady-state approximations for large-scale queues.

Lyapunov conditions guarantee uniform convergence across system scales.

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.