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Shock models with underlying birth process

Published online by Cambridge University Press:  14 July 2016

M. S. A-Hameed  and
F. Proschan
M. S. A-Hameed
Affiliation:
Florida State University
F. Proschan
Affiliation:
Florida State University
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Abstract

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process: given k shocks have occurred in [0, t], the probability of a shock occurring in (t, t + Δ] is λ kλ (t)Δ + o (Δ). We show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the ‘new better than used' property, etc.) are obtained under appropriate assumptions on λ k, λ (t), and on the probability of surviving a given number of shocks.

Keywords

SHOCK MODELSBIRTH PROCESSTOTAL POSITIVITYLIFE DISTRIBUTIONSINCREASING FAILURE RATENEW BETTER THAN USEDVARIATION DIMINISHING PROPERTYRELIABILITY

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975 , pp. 18 - 28
Copyright
Copyright © Applied Probability Trust 1975 

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References

A-Hameed, M. S. and Proschan, F. (1973) Nonstationary shock models. Stoch. Proc, Applic. 1, 383–404.Google Scholar
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