The solution of certain two-dimensional Markov models

Abstract

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.