Abstract
Many large engineering systems can be viewed (or imbedded) as a series system in time. In this paper, we introduce the structure of a repairable system and the reliabilities of these large systems are studied systematically by studying the ergodicities of certain non-homogeneous Markov chains. It shows that if the failure probabilities of components satisfy certain conditions, then the reliability of the large system is approximately exp (-β) for some β>0. In particular, we demonstrate how the repairable system can be used for studying the reliability of a large linearly connected system. Several practical examples of large consecutive-k-out-of-n:F systems are given to illustrate our results. The Weibull distribution is derived under our natural set-up.
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References
Aki, S. (1985). Discrete distributions of order k on a binary sequence, Ann. Inst. Statist. Math., 37, 205–224.
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing, Holt, Reinhard and Winston, New York.
Bollinger, R. C. (1982). Direct computation for consecutive-k-out-of-n:F system, IEEE Trans. Reliability, R-31, 444–446.
Chao, M. T. and Lin, G. D. (1984). Economical design of large consecutive-k-out-of-n:F system, IEEE Trans. Reliability, R-33, 411–413.
Chiang, D. T. and Niu, S. C. (1981). Reliability of consecutive-k-out-of-n:F system, IEEE Trans. Reliability, R-33, 411–413.
Chrysaphinou, O. and Papastavridis, S. (1988). Asymptotic distribution of a consecutive-k-out-of-n:F system, Tech. Report, University of Athens, Greece.
Fu, J. C. (1985). Reliability of a large consecutive-k-out-of-n:F system, IEEE Trans. Reliability, R-34, 127–130.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n:F system with (k−1)-step Markov dependence, IEEE Trans. Reliability, R-35, 602–606.
Fu, J. C. and Hu, B. (1987). On reliability of a large consecutive-k-out-of-n:F system with (k−1)-step Markov dependence, IEEE Trans. Reliability, R-36, 75–77.
Hwang, F. K. (1986). Simplified reliabilities for consecutive-k-out-of-n:F system, SIAM J. Algebraic Discrete Methods, 7, 258–264.
Lopez, A. (1961). Problems in stable population theory, Tech. Report, Office of Populations Research, Princeton University, Princeton.
Papastavridis, S. (1987). A limit theorem for the reliability of a consecutive-k-out-of-n:F system, Adv. in Appl. Probab., 19, 746–748.
Papastavridis, S. and Lambiris, M. (1987). Reliability of a consecutive-k-out-of-n:F system for Markov-dependent components, IEEE Trans. Reliability, R-36, 78–80.
Seneta, E. (1981). Non-Negative Matrices and Markov Chains, 2nd ed., Springer, New York-Berlin.
Tong, Y. L. (1985). A rearrangement inequality for the longest run with an application to network reliability, J. Appl. Probab., 22, 386–393.
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This research work was partially supported by the National Science Council of the Republic of China.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-9216, and by the National Science Council of the Republic of China.
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Chao, M.T., Fu, J.C. A limit theorem of certain repairable systems. Ann Inst Stat Math 41, 809–818 (1989). https://doi.org/10.1007/BF00057742
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DOI: https://doi.org/10.1007/BF00057742