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Partial orderings under cumulative damage shock models

Part of: Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Franco Pellerey
Franco Pellerey*
Affiliation:
Universitá di Milano
*
* Postal address: Universitá di Milano, Dipartimento di Matematica, Via L. Cicognara 7, 20129 Milano, Italy.
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Abstract

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

Keywords

STOCHASTIC ORDERINGSSHOCK MODELSIFRPF2 DENSITYDMRLDRFRTP2 PROPERTY

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 939 - 946
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

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

References

Karlin, S. (1968) Total Positivity, Vol. 1. Stanford University Press.Google Scholar
Karlin, S. and Proschan, F. (1960) Polya type distributions of convolutions. Ann. Math. Statist 31, 721–736.CrossRefGoogle Scholar
Kochar, S. C. (1990) On preservation of some partial orderings under shock models. Adv. Appl. Prob. 22, 508–509. Correction: Adv. Appl. Prob. 25, 1013.Google Scholar
Lynch, J., Mimmack, G. and Proschan, F. (1987) Uniform stochastic order and total positivity. Canad. J. Statist. 13, 63–69.Google Scholar
Pellerey, F. and Shaked, M. (1992) Stochastic comparison of some wear processes. Technical Report, University of Arizona, Tucson, AZ 85721.Google Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1992) Optimal allocation of resources to nodes of parallel and series systems. Adv. Appl. Prob. 24, 894–914.Google Scholar
Shanthikumar, J. G. (1988) DFR property of first-passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.CrossRefGoogle Scholar
Shanthikumar, J. G., Yamazaki, G. and Sakesegawa, H. (1991) Characterization of optimal order of server in a tandem queue with blocking. Operat. Res. Letters 10, 17–22.Google Scholar
Singh, H. (1989) On partial orderings of life distributions. Naval Res. Log. 36, 103–110.3.0.CO;2-7>CrossRefGoogle Scholar
Singh, H. and Jain, K. (1989) Preservation of some partial orderings under Poisson shock models. Adv. Appl. Prob. 21, 713–716.Google Scholar