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In this paper, we consider a repairable system with minimal repairs whose number of repairs is a positive random variable with a given probability vector. Some preservation theorems and aging properties of repairable systems are established. Under the condition that at time t the system is working, a new random variable for the residual lifetime of the system is proposed. Some stochastic ordering results among the lifetimes and residual lifetimes of two systems are obtained. Similar results for coherent systems with independent components and exchangeable components were obtained in the previous literature. Copyright © 2013 John Wiley & Sons, Ltd.

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Shannon entropy is a useful criterion for measuring the uncertainty (predictability) of lifetimes of engineering systems. In this work, we provide an explicit expression for the entropy of the residual lifetime of a working used system with exactly i failed components at time t, using dynamic signature. We also present additional results on bounds and ordering properties for the proposed entropy. We find an expression for the Jensen‐Shannon (JS) divergence of the residual lifetime of a working used system, and show that the JS divergence of the system is equal to that of its dual. An improved bound for the JS divergence is also obtained. Finally, based on the proposed entropy, we introduce a criterion using which we can prefer a system. This criterion, a distribution‐free measure that only depends on the dynamic signature, ranks systems based on their closeness to extreme systems.

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