Exponential Convergence Rates for Stochastically Ordered Markov Processes with Random Initial Conditions

TL;DR

This paper derives computable exponential convergence rates for a broad class of stochastically ordered Markov processes with random initial conditions, extending previous fixed-initial-state results and illustrating with queueing models.

Contribution

It extends existing convergence rate results to include stochastic initial conditions, providing explicit bounds using moment-generating functions.

Findings

Established exponential convergence bounds for processes with random initial states.

Applied results to M/M/1 queue with rate changes, a novel setting.

Provided explicit convergence rate formulas for practical queueing scenarios.

Abstract

In this brief paper we find computable exponential convergence rates for a large class of stochastically ordered Markov processes. We extend the result of Lund, Meyn, and Tweedie (1996), who found exponential convergence rates for stochastically ordered Markov processes starting from a fixed initial state, by allowing for a random initial condition that is also stochastically ordered. Our bounds are formulated in terms of moment-generating functions of hitting times. To illustrate our result, we find an explicit exponential convergence rate for an M/M/1 queue beginning in equilibrium and then experiencing a change in its arrival or departure rates, a setting which has not been studied to our knowledge.