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Strong stochastic convexity: closure properties and applications

Published online by Cambridge University Press:  14 July 2016

J. George Shanthikumar  and
David D. Yao
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
David D. Yao*
Affiliation:
Columbia University
*
∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
∗∗Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027–6699, USA.
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Abstract

A family of random variables {X(θ)} parameterized by the parameter θ satisfies stochastic convexity (SCX) if and only if for any increasing and convex function f(x), Ef[X(θ)] is convex in θ. This definition, however, has a major drawback for the lack of certain important closure properties. In this paper we establish the notion of strong stochastic convexity (SSCX), which implies SCX. We demonstrate that SSCX is a property enjoyed by a wide range of random variables. We also show that SSCX is preserved under random mixture, random summation, and any increasing and convex operations that are applied to a set of independent random variables. These closure properties greatly facilitate the study of parametric convexity of many stochastic systems. Applications to GI/G/1 queues, tandem and cyclic queues, and tree-like networks are discussed. We also demonstrate the application of SSCX in bounding the performance of certain systems.

Keywords

GI/G/1 QUEUESQUEUEING NETWORKSBOUNDSCONVEX ORDERINGDESIGN AND OPTIMIZATION OF QUEUEING SYSTEMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991 , pp. 131 - 145
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

J. George Shanthikumar is supported in part by NSF under grant ECS-8811234. The major part of this research was undertaken when David D. Yao was affiliated with the Division of Applied Sciences, Harvard University. He has been supported in part by NSF under grant ECS-8803183, and by ONR under contract N00014–84-K-0465.

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