Importance Measures in Multistate Systems Applications of generalisations of the Birnbaum-measure

Abstract

Several examples of use of four importance measures for repairable multistate systems. The measures are based on the classical Birnbaum reliability importance measure, and can be characterised with respect to two features. First, they are either forward-looking or backward-looking. Second, they are either measuring criticality by the probability of criticality, or by the expected impact on the system state. We consider both periodic life cycles and more stochastic life cycles, modelled by semi-Markov processes. We look at several examples and see that the rankings of the components may differ between the measures. We find a result that shows that a component that is in parallel with the rest of the system is always with importance equal to 1 and thus most important in two of the measures. We also discover a new criterion to make two and two measures equal in pairs. Also, two generalisations of the Birnbaum structural importance measure to the case of multistate systems are proposed, where the two are based on the feature of being either forward-looking or backward-looking, and we prove that the parallel theorem still holds.