Computational Methods for Reliability and Importance Measures of Weighted-Consecutive-System

We consider the two reliability systems, namely, the weighted-consecutive- k-out-of- n:F (C
^ω
(k,n:F)), and weighted-m-consecutive- k-out-of- n:F system (C
_m
^ω
(k,n:F)). The weighted-systems have in general n components, each one having a positive integer weight ω
_i
, i=1,2,...n such that total weight of all components of the system is ω=Σ
_i=1
ⁿ
ω
_i
. The C
^ω
(k,n:F) fails iff the total weight of failed consecutive components is at least k. We propose C
_m
^ω
(k,n:F) as a generalization of C
^ω
(k,n:F) which fails iff there are at least m non-overlapping groups of consecutive failed components with a total weight of at least k. Here we study the reliability, Birnbaum reliability importance, and improvement potential importance of a weighted-consecutive system based on the distribution of the failure run statistic in the sequence of weighted Bernoulli trials. We develop a simplified, efficient formula for the evaluation of reliability and importance measures of the systems under consideration, and demonstrate the results numerically.