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On the signature of a system under minimal repair

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The signature of a coherent system with independent and identically distributed component lifetimes is a useful tool in the study and comparison of lifetimes of systems. The signature of a coherent system with n components is a vector whose kth element is the probability that the kth component failure is fatal to the system and is a distribution-free measure of the design of the system. Recently, the notion of dynamic signature was introduced in the literature. Here, one considers a system that is inspected at some time t and found to be working, with i, say, failed components. The dynamic signature is then the signature of the system consisting of the remaining n-i components. In the present paper, we consider the situation when a system failure is observed, and we assume that the only information available is the time of failure, t, and the number of failed components, k, at failure. A minimal repair is assumed. The paper presents a construction of the probabilistic mechanism of the failure process following the failure and minimal repair, conditioning on the available information. The signature of the remaining system after repair is called the conditional dynamic signature, which is similar to the dynamic signature, and the two can be computed from the same set of residual signatures. It is conjectured that the dynamic signature generally dominates stochastically the conditional dynamic signature under comparable scenarios. A condition that guarantees this domination is presented. The condition is shown to be sufficient but not necessary. Copyright (c) 2014 John Wiley Sons, Ltd.

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