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On the Number of Component Failures in Systems Whose Component Lives Are Exchangeable
Sheldon M. Ross, Mehrdad Shahshahani and Gideon Weiss
Mathematics of Operations Research
Vol. 5, No. 3 (Aug., 1980), pp. 358-365
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3689442
Page Count: 8
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We consider a system that is composed of finitely many independent components each of which is either "on" or "off" at any time. The components are initially on and they have common on-time distributions. Once a component goes off, it remains off foreover. The system is monotone in the sense that if the system is off whenever each component in a subset S (called a cut set) of components is off, then that is also true for every subset of components containing S. We are interested in studying the properties of N, the number of components that are off at the moment the system goes off. We compute the factorial moments of N in terms of the reliability function. We also prove that N is an increasing failure rate average random variable and present a duality result. We consider the special structure in which the minimal cuts sets do not overlap and we prove a conjecture of El-Neweihi, Proschan and Sethuraman which states that N is an increasing failure rate random variable. Then we consider the special case of nonoverlapping minimal path sets, and in the final section we present an application to a shock model.
Mathematics of Operations Research © 1980 INFORMS