New Results on the Barlow–Proschan and Natvig Measures of Component Importance in Nonrepairable and Repairable Systems

In this paper dynamic and stationary measures of importance of a component in a binary system are considered. To arrive at explicit results we assume the performance processes of the components to be independent and the system to be coherent. Especially, the Barlow–Proschan and the Natvig measures are treated in detail and a series of new results and approaches are given. For the case of components not undergoing repair it is shown that both measures are sensible. Reasonable measures of component importance for repairable systems represent a challenge. A basic idea here is also to take a so-called dual term into account. According to the extended Barlow–Proschan measure a component is important if there are high probabilities both that its failure is the cause of system failure and that its repair is the cause of system repair. Even with this extension results for the stationary Barlow–Proschan measure are not satisfactory. According to the extended Natvig measure a component is important if both by failing it strongly reduces the expected system uptime and by being repaired it strongly reduces the expected system downtime. With this extension the results for the stationary Natvig measure seem very sensible.