Patterns of the Birnbaum importance in linear consecutive--out-of- systems

The Birnbaum importance is a well-known measure that evaluates the relative contribution of components to system reliability. There exist certain patterns of the component Birnbaum importance (i.e., the relative order of the Birnbaum importance values to the individual components) for linear consecutive-k-out-of-n (Lin/Con/k/n) systems when all components have the same reliability p. Previous research has shown that based on the Birnbaum importance, plausible patterns and conjectures exist. This article summarizes and annotates the Birnbaum importance patterns for Lin/Con/k/n systems, proves new Birmbaum importance patterns conditioned on the value of p, disproves some patterns that were conjectured or claimed in the literature, and makes new conjectures based on comprehensive computational tests and analysis. More important, this article defines a concept of segment in Lin/Con/k/n systems for analyzing the Birnbaum importance patterns and investigates the relationship between the Birnbaum importance and the common component reliability p and the relationship between the Birnbaum importance and the system size n. One can then use these relations to further understand the proved, disproved, and conjectured Birnbaum importance patterns.