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New better than used processes

Published online by Cambridge University Press:  01 July 2016

Albert W. Marshall  and
Moshe Shared
Albert W. Marshall*
Affiliation:
University of British Columbia
Moshe Shared*
Affiliation:
University of Arizona
*
∗Postal address: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Y4.
∗∗Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, U.S.A.
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Abstract

A stochastic process , such that P{Z(0) = 0} = 1, is said to be new better than used (NBU) if, for every x, the first-passage time Tx = inf {t: Z(t) > x} satisfies P{TX > s + t} for everys . In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver–Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.

Keywords

RELIABILITY THEORYQUEUESDAMSINVENTORYSHOCK MODELSGENERAL RELEASE OR RECOVERY RULESWIDE AND SHORT GAPS IN A RENEWAL PROCESSCLUSTERS OF RENEWAL EVENTSRANDOM REPAIR-TIMES PROCESSESGAVER–MILLER PROCESSESMARKOV PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 3 , September 1983 , pp. 601 - 615
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Supported in part by the National Science Foundation at Stanford University, and in part by the National Sciences and Engineering Research Council, Canada.

Supported in part by National Science Foundation Grant MCS-79-27150.

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