Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-09T20:05:20.856Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brémaud, P. (1981). Point processes and queues. New York: Springer-Verlag.Google Scholar
Daley, D.J. (1968). Stochastically monotone Markov chains. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandle Gebiete 10: 305–3 17.Google Scholar
Daley, D.J. & Rolski, T. (1984). Some comparability results for waiting times in single- and many-server queues. Journal of Applied Probability 21: 887–900.CrossRefGoogle Scholar
Fox, B. (1966). Discrete optimization via marginal analysis. Management Science 13: 210–216.CrossRefGoogle Scholar
Kamae, T., Krengel, U., & O'Brien, G. (1977). Stochastic inequalities on partially ordered spaces. Annals of Probability 5: 899–912.Google Scholar
Keilson, J. (1979). Markov chain models–Rarity and exponentialiry. New York: Springer- Verlag.Google Scholar
Massey, W. A. (1987). Stochastic ordering for Markov processes. Mathematics of Operations Research 12: 350–367.Google Scholar
Melamed, B. & Walrand, J. (1986). On the one-dimensional distributions of counting processes with stochastic intensities. Stochastics 19: 1–9.Google Scholar
Ross, S. M. (1983). Introduction to stochastic dynamic programming. New York: Academic Press.Google Scholar
Ross, S.M. & Schechner, Z. (1984). Some reliability applications of the variability ordering. Operations Research 32: 679–687.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1987). Temporal stochastic convexity and concavity. Stochastic Processes and Their Applications 27: 1–20.Google Scholar
Shaked, M. & Shanthikumar, J. G. (1988). Stochastic convexity and its applications. Advances in Applied Probability 20: 427–446.CrossRefGoogle Scholar
Shaked, M. & Shanthikumar, J.G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509–531.Google Scholar
Shanthikumar, J.G. & Yao, D.D. (1987). Optimal server allocation in a system of multi-server stations. Management Science 33: 1173–1180.CrossRefGoogle Scholar
Shanthikumar, J.G. & Yao, D.D. (1988). Second-order properties of the throughput of a closed queueing network. Mathematics of Operations Research 13: 524–534.Google Scholar
Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York:Wiley.Google Scholar
Whitt, W. (1981). Comparing counting processes and queues. Advances in Applied Probability 13: 207–220.Google Scholar
Whitt, W. (1986). Stochastic comparisons for non-Markov processes. Mathematics of Operations Research 11: 608–618.Google Scholar