On the Kullback Leibler information for mixed systems

Entropy and Kullback–Leibler (KL) information for engineering systems have been studied in both statistical and reliability contexts. In this paper, we prove that the KL information between distributions of mixed system lifetimes and the corresponding component lifetimes and also the associated order statistics are distribution free and depends only on the signature of the system provided that lifetimes of components are independent and identically distributed (iid). The obtained results are used to find the closest and the farthest distribution of order statistics from the distribution of the system’s lifetime which is useful to approximate stochastic behaviour of mixed systems when the number of components is large. Finally, we provide bounds and also use the results to obtain a more preferable system among all systems. Some illustrative examples are also given.

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