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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

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