Abstract
We suggest four new measures of importance for repairable multistate systems based on the classical Birnbaum measure. Periodic component life cycles and general semi-Markov processes are considered. Similar to the Birnbaum measure, the proposed measures are generic in the sense that they only depend on the probabilistic properties of the components and the system structure. The multistate system model encodes physical properties of the components and the system directly into the structure function. As a result, calculating importance is easy, especially in the asymptotic case. Moreover, the proposed measures are composite measures, combining importance for all component states into a unified quantity. This simplifies ranking of the components with respect to importance. The proposed measures can be characterized with respect to two features: forward-looking versus backward-looking and with respect to how criticality is measured. Forward-looking importance measures focus on the next component states, while backward-looking importance measures focus on the previous component states. Two approaches to measuring criticality are considered: probability of criticality versus expected impact. Examples show that the different importance measures may result in unequal rankings.
Highlights
A main problem in reliability theory is to determine how the reliability of a complex system can be determined from knowledge of the reliabilities of its components
In the present paper we focus on generic importance measures
Forward-looking importance measures focus on the component states
Summary
A main problem in reliability theory is to determine how the reliability of a complex system can be determined from knowledge of the reliabilities of its components. The paper introduced a reliability importance vector where the ith entry of the vector represented the impact on the system reliability given that the component was improved from state i − 1 to state i This approach can be viewed as a natural extension of the Birnbaum measure to the multistate case. Si et al (2013) extend the integrated importance measure to estimate the effect of a component residing at certain states on the performance of entire multistate systems. In order to facilitate this, the modelling framework must be extended to include additional elements such as objective functions, cost functions and demand models This is a very interesting development which makes importance measures applicable to a much wider range of problems. The semi-Markov models used in this paper are similar to the ones considered in Dui et al (2015)
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