Exponential ergodicity and steady-state approximations for a class of Markov processes under fast regime switching

TL;DR

This paper investigates the ergodic behavior and steady-state diffusion approximations of Markov-modulated birth-death processes under fast regime switching, establishing conditions for exponential ergodicity and convergence rates.

Contribution

It provides new theoretical results linking ergodic properties of scaled processes to the original process and introduces convergence rate analysis for steady-state diffusion approximations.

Findings

Exponential ergodicity of the averaged process implies ergodicity of the original process.

Convergence rates for moments of diffusion approximations are established.

Several examples demonstrate practical applications of the theory.

Abstract

We study ergodic properties of a class of Markov-modulated general birth-death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the 'averaged' fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov modulated birth-death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.