Computable exponential convergence rates for stochastically ordered Markov processes

Abstract

Let ${\Phi_t, t \geq 0}$ be a Markov process on the state space $[0, \infty)$ that is stochastically ordered in its initial state. Examples of such processes include server workloads in queues, birth-and-death processes, storage and insurance risk processes and reflected diffusions. We consider the existence of a limiting probability measure $\pi$ and an exponential "convergence rate" $\alpha > 0$ such that $$\lim_{t \to \infty} e^{\alpha t} \sup_A |P_x[\Phi_t \epsilon A] - \pi (A)| = 0$$ for every initial state $\Phi_0 \equiv x$.

The goal of this paper is to identify the largest exponential convergence rate $\alpha$, or at least to find computationally reasonable bounds for such a "best" $\alpha$. Coupling techniques are used to derive such results in terms of (i) the moment-generating function of the first passage time into state ${0}$ and (ii) solutions to drift inequalities involving the generator of the process. The results give explicit bounds for total variation convergence of the process; convergence rates for $E_x [f(\Phi_t)]$ to $\int f(y) \pi (dy)$ for an unbounded function f are also found. We prove that frequently the bounds obtained are the best possible. Applications are given to dam models and queues where first passage time distributions are tractable, and to one-dimensional reflected diffusions where the generator is the more appropriate tool. An extension of the results to a multivariate setting and an analysis of a tandem queue are also included.