The monotonicity of component importance measures in linear consecutive-k-out-of-n systems
The "monotonicity" behavior of component importance measures in linear consecutive-k-out-of-n systems is studied through the formulation of a number of importance measures for these systems when all components are assumed to be independently and equally reliable. For the system with unequally reliable components, specific formulations are also developed. The structure and reliability functions for linear consecutive-k-out-of-n systems are derived through the notion of minimal-path vector representation of system structure;Three types of component importance measures are considered in this study: first, importance measures based on critical vectors which include the Birnbaum and Barlow-Proschan measures; second, importance measures based on minimal-path vectors, which include the Deegan-Packel measure; and third, importance measures based on minimal-cut vectors, which include the Vesely-Fussel measure. It is shown that, for k ≤ n ≤ 2k, these measures are always "monotone." However, for n > 2k, only the Deegan-Packel and Vesely-Fussel measures are "monotone," whereas the Birnbaum and Barlow-Proschan measures are shown to be "nonmonotone" for the entire domain of component reliability p, where n is in the range (2k + 1,3k + 1). For n > 3k + 1, the Birnbaum measure is "nonmonotone" for certain p and k, and the Barlow-Proschan measure is "nonmonotone" when n is sufficiently large.