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Shock models by underlying counting process

Part of: Special processes Distribution theory - Probability

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.

References

Abouammoh, A. M., Hindi, ?. I. and Ahmed, A. N. (1988) Shock models with NBUFR and NBAFR survivals. Trab. Estadist. 3, 97–113.Google Scholar
A-Hameed, ?. S. and Proschan, F. (1975) Shock models with underlying birth process. J. Appl. Prob. 12, 18–28.Google Scholar
Barlow, R. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing; Probability Models. To Begin With, Silver Spring.Google Scholar
Block, H. W. and Savits, T. H. (1978) Shock models with NBUE survival. J. Appl. Prob. 15, 621–628.CrossRefGoogle Scholar
Bryson, M. C. and Siddiqui, M. M. (1969) Some criteria for aging. J. Amer. Statist. Assoc. 64, 1472–1483.Google Scholar
Cao, J. and Wang, Y. (1991) The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28, 473–479.CrossRefGoogle Scholar
Deshpande, J. V., Kochar, S. C. and Singh, H. (1986) Aspects of positive ageing. J. Appl. Prob. 23, 473–479.Google Scholar
Fagiuoli, E. and Pellerey, F. (1994) Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31(2).CrossRefGoogle Scholar
Khalique, A. (1989) On discrete failure-time distributions. Rel. Eng. Syst. Safety 25, 99–107.CrossRefGoogle Scholar
Klefsjö, B. (1981) HNBUE survival under some shock models. Scand. J. Statist. 8, 39–47.Google Scholar
Klefsjö, B. (1982) The HNBUE and HNWUE classes of life distributions. Naval Res. Logist. Quart. 29, 331–344.CrossRefGoogle Scholar
Loh, W. (1984) A new generalization of the class of NBU distributions. IEEE Trans. Reliability 33, 419–422.CrossRefGoogle Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar