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Spatiotemporal Convexity of Stochastic Processes and Applications

Published online by Cambridge University Press:  27 July 2009

J. George Shanthikumar  and
David D. Yao
J. George Shanthikumar
Affiliation:
School of Business Administration University of California, Berkeley, California 94720
David D. Yao
Affiliation:
Department of Industrial Engineering and Operations Research Columbia University, New York, New York 10027-6699
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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.

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Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 1 , January 1992 , pp. 1 - 16
Copyright
Copyright © Cambridge University Press 1992

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