Birth-Death Processes in Interactive Random Environments

Abstract

We study birth-death processes in interactive random environments, where not only the birth and death rates depend on the state of the environment, but also the evolution dynamics of the environment depends on the state of the birth-death process. Two types of random environments are considered, either pure jump (finite or infinite countable) Markov process or a reflected (jump) diffusion process. The joint Markov process is constructed in such a way that the interactive impact is explicitly characterized, and the joint invariant measure takes an explicit form which may be regarded as a type of product form. We provide a few examples in queueing (multi-server and infinite-server models) and population growth models, to illustrate the construction of the joint Markov process and the conditions under which the explicit product-form invariant measure can be derived.We then study the rate of convergence to stationarity of these models. We consider two settings which lead to either an exponential rate or a polynomial rate. In both settings, we assume that the underlying environmental Markov process has an exponential rate of convergence, however, the convergence rates of the joint Markov process depend on the certain conditions on the birth and death rates. We use a coupling approach to prove these results on the rates of convergence.Key words and phrases. birth-death processes, interactive random environment, Markov jump process, (jump) diffusion process, invariant measures, product-form formula, exponential/polynomial rate of convergence to stationarity.