  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.
