Birth and Death Processes in Interactive Random Environments

TL;DR

This paper investigates birth and death processes in interactive random environments, analyzing their invariant measures and convergence rates, with applications to queueing and population-growth models.

Contribution

It introduces models of birth-death processes with interactive environments and derives explicit invariant measures and convergence conditions.

Findings

Explicit invariant measures for the models are derived.

Conditions for exponential and polynomial convergence rates are established.

Coupling methods are used to analyze convergence to stationarity.

Abstract

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results a coupling method turns out to be useful.