Stochastic Formulations of Importance Measures and Their Extensions to Multi-State Systems

In the case of binary state reliability system, importance measures of components consisting a system, such as structural importance measure, Birnbaum importance measure, criticality importance measures and some other importance measures, have been proposed and used effectively for practical risk and safety problems. When we want to judge which component or factor should be maintained first to improve the system’s performance, the importance measure suggests us that the most important component in a sense of an importance measure should be the first candidate. But these measures are not necessarily defined clearly from a stochastic theoretical point of view, and also extensions of them to the case of multi-state systems are not sufficiently achieved. In this paper, we show stochastically clear definitions of these importance measures with basic mathematical ideas behind them. The definitions do not need a usually assumed stochastic independence among components and then we may naturally extend the measures for the binary case to the multi-state case in various ways. An algorithm to give an extended Birnbaum importance measure from minimal state vectors which uniquely determine the structure function of the system is also given. And a calculation method of importance measures via a modular decomposition is also given.