Efficient algorithm for reliability and importance measures of linear weighted-(n,f,k) and ⟨n,f, k⟩ systems

A weighted-(n,f,k): F/G < n,f, k >: F/G) system consists of n components ordered in a line or circle and the system fails/works if and only if the total weight of failed/working components is at least f or (and) total weight of consecutive failed/working components is at least k. In this paper, we study the reliability and probability based reliability importance measures for linear weighted-(n,f, k): F and < n,f, k >: F system through a joint distribution of weighted failure-run-statistics in the sequence of Weighted Markov Binary Trials. Through the joint distributions studied, the reliability and reliability importance measures of f-out-of-n: F, consecutive-k-out-of-n: F, (n,f,k): F and < n, f, k >: F systems and their weighted versions (in all eight systems) can also be obtained. We also bring out the inter-relationships between reliabilities and Birnbaum importance of the weighted-f-out-of-n: F, weighted-consecutive-k-out-of-n: F, weighted-(n,f, k): F and < n,f,k >: F systems. Further, we demonstrate the results developed numerically. Our formula for reliability of weighted-(n,f, k): F system is more efficient in terms of cpu time than the existing method. (C) 2017 Elsevier Ltd. All rights reserved.