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Convexity of a set of stochastically ordered random variables

Published online by Cambridge University Press:  01 July 2016

Moshe Shaked  and
J. George Shanthikumar
Moshe Shaked*
Affiliation:
University of Arizona
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗Postal address: Department of Mathematics Building # 89, The University of Arizona, Tucson, AZ 85721, USA.
∗∗Postal address: School of Business Administration, 350 Barrows Hall, University of California, Berkeley, CA94720, USA.
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Abstract

It is shown that a set of random variables with increasing and convex (concave) survival functions are stochastically increasing and convex (concave) in the sample path sense. This stochastic convexity (concavity) result is then used to establish convexity (concavity) results for (i) a single-server queueing system with a time-out control policy, (ii) residual life, (iii) stationary renewal excess life and (iv) M/G/1 queues. These results are new and could not be derived without the direct or indirect aid of the above stochastic convexity (concavity) result. Furthermore, we illustrate that the above stochastic convexity (concavity) result can be applied to obtain new bounds for queueing systems. Specifically, let be the waiting time of the nth customer in a GI/G/1 queue with inter-arrival time survival function and service time survival function . Using the above convexity result it is shown that if and for some such that then for all increasing convex functions φ, whenever the expectations exist. A similar result for and is also obtained. Other examples are also included.

Keywords

STOCHASTIC CONVEXITYSTOCHASTIC ORDERINGGI/G/1 QUEUECONTROLLED ARRIVAL QUEUERESIDUAL LIFESTATIONARY RENEWAL EXCESS LIFE
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 160 - 177
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

Supported by the Air Force Office of Scientific Research, USAF under Grant AFOSR-84–0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.

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