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On the Failure Rates of Consecutive−k−out−of−n Systems

Published online by Cambridge University Press:  27 July 2009

F.K. Hwang  and
Y.C. Yao
F.K. Hwang
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey 07974
Y.C. Yao
Affiliation:
Colorado State University, Fort Collins, Colorado 80523
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Extract

It is known that the lifetime of a k−out−of−n system has increasing failure rate (IFR) if all of its components have independent and identically distributed IFR lifetimes. Derman, Lieberman, and Ross raised the same question for consecutive−k−out−of−n systems. But the scarcity of results gives no clue as to whether most of such systems have IFR's. In this paper, we completely solve the k = 2 and k = 3 cases. We prove that these systems, with a handful of exceptions, do not have IFR's (contradicting the result of Derman, Lieberman, and Ross that a consecutive-2-out-of-n cycle has IFR for any n). Our result suggests the conjecture that for every fixed k there exists nk such that for every n ≧ nk a consecutive-k−out−of−n system does not have IFR. We also prove that for every fixed d there exists nd large enough such that for all n ≧ nd a consecuive.(n – d)−out−of−n system has IFR. In particular, if the system is a line, then nd = 3d +1 for d ≧ 1.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 4 , Issue 1 , January 1990 , pp. 57 - 71
Copyright
Copyright © Cambridge University Press 1990

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References

Bollinger, R.C. (1982). Direct computation for consecutive−k−out−of−n: F systems. IEEE Transactions on Reliability R-31: 444–446.CrossRefGoogle Scholar
Chen, R.W. & Hwang, F.K. (1985). Failure distribution of consecutive−k−out−of−n: F systems. IEEE Transactions on Reliability R-34:338–341.CrossRefGoogle Scholar
Chiang, D. & Niu, S.C. (1981). Reliability of consecutive−k−out−of−n: F system. IEEE Transactions on Reliability R-30: 87–89.CrossRefGoogle Scholar
Derman, C., Lieberman, G.J., & Ross, S.M. (1982). On the consecutive−k−out−of−n: F system. IEEE Transactions on Reliability R-31: 57–63.CrossRefGoogle Scholar
Hwang, F.K. (1986). Simplified reliabilities for consecutive−k−out−of−n systems. SIAM Journal of Algebraic and Discrete Methods 7: 258–264.CrossRefGoogle Scholar
Hwang, F.K. & Yao, Y.C. A direct argument for Kaplansky's theorem on cyclic arrangement and its generalization. Operations Research Letters (in press).Google Scholar
Kaplansky, I. (1943). Solution of the “problèm des ménages.” Bulletin of the American Mathematical Society 49: 784–785.CrossRefGoogle Scholar
Lambiris, M. & Papastravridis, S. (1985). Exact probability formulas for linear and circular consecutive−k−out−of−n: F systems. IEEE Transactions on Reliability R-34: 124–126.CrossRefGoogle Scholar
Shanthikumar, J.G. (1985). Lifetime distributions of consecutive−k−out−of−n: F systems with exchangeable lifetime. IEEE Transactions on Reliability R-34: 480–483.CrossRefGoogle Scholar