Joint Structural Importance in Consecutive-k Systems

The joint structural importance (JSI) is an important measure of how two components interact in contributing to the system reliability. The value of JSI is positive (negative) if and only if one component becomes more important (less important) when the other works. A consecutive-k-out-of-n system is a linear arrangement of n components such that the system is failed if and only if some consecutive k components are all failed. In this paper, we study joint structural importance ) , ( j i JSI in the consecutive-k-out-of-n system. We completely solve ) , ( j i JSI for 1 = k (the series system), n k = (the parallel system), 1 − = n k , and 2 − = n k , respectively. For the other k , we prove that < + = < < = ′ ) 2 , 1 ( ) , 1 ( 0 ) , 1 ( ) , 1 ( k JSI n JSI k JSI j JSI ) 1 , 1 ( ) , 1 ( + < k JSI j JSI , for 1 2 − ≤ ′ ≤ k j and 1 3 − ≤ ≤ + n j k . For a fixed i , we prove that the graph of ) , ( j i JSI has a W-shape property for } 1 , min{ } 1 , 1 max{ + + ≤ ≤ − − k i n j k i with 0 ) , ( = i i JSI . We also present exact formula for ) , ( j i JSI and obtain many relations among them. Key-Words: Joint structural importance, Consecutive-k-out-of-n system, Reliability, Birnbaum importance.