Computable exponential convergence rates for stochastically ordered Markov processes

Abstract

Let ${\\Phi_t, t \\geq 0}$ be a Markov process on the state space $[0,\n\\infty)$ that is stochastically ordered in its initial state. Examples of such\nprocesses include server workloads in queues, birth-and-death processes,\nstorage and insurance risk processes and reflected diffusions. We consider the\nexistence of a limiting probability measure $\\pi$ and an exponential\n"convergence rate" $\\alpha > 0$ such that $$\\lim_{t \\to \\infty}\ne^{\\alpha t} \\sup_A |P_x[\\Phi_t \\epsilon A] - \\pi (A)| = 0$$ for every initial\nstate $\\Phi_0 \\equiv x$.\n¶ The goal of this paper is to identify the largest exponential\nconvergence rate $\\alpha$, or at least to find computationally reasonable\nbounds for such a "best" $\\alpha$. Coupling techniques are used to\nderive such results in terms of (i) the moment-generating function of the first\npassage time into state ${0}$ and (ii) solutions to drift inequalities\ninvolving the generator of the process. The results give explicit bounds for\ntotal variation convergence of the process; convergence rates for $E_x\n[f(\\Phi_t)]$ to $\\int f(y) \\pi (dy)$ for an unbounded function f are\nalso found. We prove that frequently the bounds obtained are the best possible.\nApplications are given to dam models and queues where first passage time\ndistributions are tractable, and to one-dimensional reflected diffusions where\nthe generator is the more appropriate tool. An extension of the results to a\nmultivariate setting and an analysis of a tandem queue are also\nincluded.