Birnbaum importance measure of network based on C-spectrum under saturated poisson distribution

Importance measures usually provide numerical indicator to decide which component is more important for network reliability improvement or more critical for network failure. The concept of C-spectrum is a useful tool to implement the importance measures for network, which solely depends on network structure. In this paper, we analyze a network that consists of n components (edges). Under the condition that the distribution of the number of failed edges is given, the properties of traditional Birnbaum importance measure (BIM) are generalized and investigated. First, we derive a formula for BIM based on C-spectrum and establish a sufficient and necessary condition for comparing two edges according to their BIMs. Secondly, under the special case where the number of failed edges follows a saturated Poisson distribution with intensity λ, for enough small λ the BIM ranking is structural ranking, i.e., depending solely on the network structure through the C-spectrum. Finally, an example is presented to explain how we can rank edges according to their BIMs.