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

•Reliability & importance measures for weighted-(n,f,k) & 〈n,f,k〉 system are studied.•The basis of the study is joint probability distribution of weighted run statistic.•Algorithm is more efficient in terms of cpu time than one of the existing methods.

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.