Spatiotemporal Convexity of Stochastic Processes and Applications

Abstract

A stochastic process {Xt(s)} is viewed as a collection of random variables parameterized by time (t) and the initial state (s). {Xt(s)} is termed spatiotemporally increasing and convex if, in a sample-path sense, it is increasing in s and t and satisfies a directional convexity property, which implies that it is increasing and convex in s and in t (individually) and is supermodular in (s, t). Simple sufficient conditions are established for a uniforniizable Markov process to be spatiotemporally increasing and convex. The results are applied to study the convex orderings in GI/M(n)/l and M(n)/G/1 queues and to solve the optimal allocation of a joint setup among several production facilities. For a counting process that possesses a stochastic intensity, we show that its spatiotemporal behavior can be characterized by its conditional intensity via a birth process.