Logic Differential Calculus in Time-Dependent Importance Analysis Based on Minimal Cut Vectors

Evaluation of influence of system components on operation of a system is an important task in reliability analysis. It is addressed in its special part known as importance analysis. This analysis uses various measures for this purpose. Some of the most useful are Birnbaum's importance and Fussell-Vesely's Importance (FVI). The former is based on quantification of circumstances under which a failure/repair of a system is caused by a failure/repair of a given component, while the latter focuses on minimal scenarios that result in a repair of a nonfunctioning system. These scenarios are commonly known as minimal cut vectors. In this paper, we present how time-dependent FVI can be computed without a priori knowledge of these scenarios. Logic differential calculus together with a theory of stochastic processes is used for this task.