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Shock models by underlying counting process

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Franco Pellerey
Franco Pellerey*
Affiliation:
Universitá di Milano
*
∗ Present address: Via Castellamonte 11, 10010 Banchette (TO), Italy.
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Abstract

Suppose that a device is subjected to shocks governed by a counting process N = {N(t), t ≧0}. The probability that the device survives beyond time t is then H̄(t)=Σk=0∞Q̄ℙ[N(t)=k], where Q̄k is the probability of surviving k shocks. It is known that H is NBU if the interarrivals Uk, ∊ ℕ+, are independent and NBU, and Q̄k+j ≦ Q̄k· Q̄j holds whenever k, j ∊ ℕ. Similar results hold for the classes of the NBUE and HNBUE distributions. We show that some other ageing classes have similar properties.

Keywords

SHOCK MODELSNBUNBUEHNBUENBU(2)NBUCNBUFRNBAFR

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60K05: Renewal theory 60E05: Distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 156 - 166
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research carried out while the author was visiting University of Arizona, Tucson, AZ.

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