Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-11T19:54:18.519Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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 

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.)

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.

References

Cooper, R. B. (1981) Introduction to Queueing Theory, 2nd edn. North-Holland, New York.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277289.Google Scholar
Meester, L. E. and Shanthikumar, J. G. (1990) Concavity of the throughput of tandem queueing systems with finite buffer storage space. Adv. Appl. Prob. 22, 764–767.Google Scholar
Price, T. G. (1974) Probability Models of Multiprogrammed Computer Systems. Ph.D. Dissertation, Stanford University.Google Scholar
Rüschendorf, L. (1983) Solution of a statistical optimization problem by rearrangement methods. Metrika 30, 55–61.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1988) Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1990a) Convexity of a set of stochastically ordered random variables. Adv. Appl. Prob. 22, 160–167.Google Scholar
Shared, M. and Shanthikumar, J. G. (1990b) Parametric stochastic convexity and concavity of stochastic processes. Ann. Inst. Statist. Math. 42, 509–531.Google Scholar
Shanthikumar, J. G. and Yao, D. D. (1987) Spatiotemporal convexity of stochastic processes and applications. Submitted for publication.Google Scholar
Stidham, S. (1974) A last word on L = ?W. Operat. Res. 20, 417–421.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
Suri, R. and Leung, Y. T. (1987) Single run optimization of a Siman model for closed loop flexible assembly systems. Proc. 1987 Winter Simulation Conf., Atlanta, GA.Google Scholar
Trivedi, K. S., Wagner, R. A. and Sigmon, T. M. (1980) Optimal selection of CPU speed, device capabilities and file assignments. J. Assoc. Comput. Mach. 27, 457–473.Google Scholar
Weber, R. R. (1983) A note on waiting time in single server queues. Operat. Res. 31, 950–951.Google Scholar