Stochastic Orderings for Markov Processes on Partially Ordered Spaces

Abstract

The purpose of this paper is to develop a unified theory of stochastic ordering for Markov processes on countable partially ordered state spaces. When such a space is not totally ordered, it can induce a wide range of stochastic orderings, none of which are equivalent to sample path comparisons. Similar comparison theorems are also developed for non-Markov processes that are functions of Markov processes and for time-inhomogeneous Markov processes. Such alternative orderings can be quite useful when analyzing multi-dimensional stochastic models such as queueing networks.