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Analysis of R-out-of-N repairable systems: the case of phase-type distributions

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Yonit Barron ,
Esther Frostig  and
Benny Levikson
Yonit Barron*
Affiliation:
University of Haifa
Esther Frostig*
Affiliation:
University of Haifa
Benny Levikson*
Affiliation:
University of Haifa
*
∗ Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel.
∗ Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel.
∗ Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel.
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Abstract

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

Keywords

Availabilityphase-type distributionMarkov renewal processregenerative pointsemi-regenerative pointup timedown time

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K15: Markov renewal processes, semi-Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 116 - 138
Copyright
Copyright © Applied Probability Trust 2004 

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References

Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.Google Scholar
Assaf, D. and Levikson, B. (1982). Closure of phase type distributions under operations arising in reliability theory. Ann. Prob. 10, 265–269.Google Scholar
Aven, T. and Jensen, U. (1998). Stochastic Models in Reliability (Appl. Math. 41). Springer, New York.Google Scholar
Çinlar, E., (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Fawzi, B. B. and Hawkes, A. G. (1991). Availability of an R-out-of-N system with spares and repairs. J. Appl. Prob. 28, 397–408.CrossRefGoogle Scholar
Frostig, E. and Levikson, B. (2002). On the availability of R out of N repairable system. Naval Res. Logistics 49, 483–498. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Neuts, M. F. (1975). Probability distributions of phase type. In Liber Amicorum Professor Emeritus H. Florin. Department of Mathematics, University of Louvain, pp. 173–206.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Neuts, M. F. and Takahashi, Y. (1981). Asymptotic behavior of the stationary distributions in the GI/PH/c queue with heterogenous servers. Z. Wahrscheinlichkeitsth. 57, 441–452.CrossRefGoogle Scholar
Neuts, M. F., Pérez-Ocón, R. and Torres-Castro, I. (2000). Repairable models with operating and repair times governed by phase type distributions. Adv. Appl. Prob. 32, 468–479.Google Scholar
Ross, S. M. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar