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Convergence of the number of failed components in a Markov system with nonidentical components

Part of: Special processes Operations research and management science Survival analysis and censored data Markov processes

Published online by Cambridge University Press:  14 July 2016

Jean-Louis Bon  and
Eugen Păltănea
Jean-Louis Bon*
Affiliation:
Université Lille-1
Eugen Păltănea
Affiliation:
Université Paris-Sud
*
∗ Postal address: Université Lille-1 EUDIL, Département GIS, Cité Scientifique, 59655 Villeneuve d'Ascq cedex, France. Email address: [email protected]
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Abstract

For most repairable systems, the number N(t) of failed components at time t appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable N(t) with N(∞), especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as Ycart (1999) but without the assumption of identical components.

Keywords

Markovreliabilityavailabilitycut-offtotal variation

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 90B25: Reliability, availability, maintenance, inspection
Secondary: 60K10: Applications (reliability, demand theory, etc.) 62N05: Reliability and life testing
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 882 - 897
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

∗∗

Current address: Universitatea Transilvania, 2200 Brasov, Romania.

References

Barbour, A. D., Chen, L. H. Y., and Loh, W.-L. (1992). Compound Poisson approximation for nonnegative random variables via Stein's method. Ann. Prob. 20, 1843–1866.CrossRefGoogle Scholar
Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA 93, 1659–1664.CrossRefGoogle ScholarPubMed
Giné, E., Grimmet, G. R., and Saloff-Coste, L. (1997). Lectures on Probability Theory and Statistics (Lecture Notes Math. 1665). Springer, Berlin.Google Scholar
Gleser, L. J. (1975). On the distribution of the number of successes in independent trials. Ann. Prob. 3, 182–188.CrossRefGoogle Scholar
Hoeffding, W. (1959). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27, 713–721.CrossRefGoogle Scholar
Păltănea, E. (2000). Analyse de fiabilité des grands systèmes réparables Markoviens stratifiés. Doctoral Thesis, Université Paris-Sud, Orsay.Google Scholar
Petrov, V. (1995). Limit Theorems in Probability Theory. Clarendon Press, Oxford.Google Scholar
Samuels, S. M. (1965). On the number of successes in independent trials. Ann. Math. Statist. 36, 1272–1278.CrossRefGoogle Scholar
Weba, N. (1999). Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities. J. Appl. Prob. 36, 97–104.CrossRefGoogle Scholar
Ycart, B. (1999). Cutoff for samples of Markov chains. ESAIM Prob. Statist. 3, 89–107.CrossRefGoogle Scholar