Reliability and Reliability Importance of Weighted-$r$ -Within-Consecutive-$k$ -out-of-$n: F$ System

In this paper, we consider a weighted-r-within-consecutive-k-out-of-n : F system. The weighted system has in general n components, each one having a positive integer weight wi, i = 1, 2, . . . , n. The weighted-r-within-consecutive-k-out-of-n : F system fails if and only if the total weight of failed components among k consecutive components is at least r. We introduce a binomial-type weighted scan statistic and study the reliability, the Birnbaum, improvement potential importance and Bayesian reliability importance of the system taken into consideration. We develop an explicit closed-form formula for the evaluation of reliability and reliability importance measures of a weighted-r-within-consecutive-k-out-of-n : F system and demonstrate the results numerically. We present a study showing the effectiveness of the method in terms of CPU time.