Abstract—Traditional importance measures seldom consider
how the number of failed components influences the network reliability. This paper proposes two importance measures under the circumstance that the failure sequence of the components follows a counting process. The first importance measure aims to assess the contribution of the individual component (edge) to the network failure. The second evaluates the contribution of the individual component to the network functionality. Both importance measures are time-dependent functions, and their values are jointly determined by the network structure and the distribution of the number of failed components at a particular time. We prove that the proposed importance measures are able to generate consistent rankings based on edge’s impact on the network reliability behavior. When networks possess special structure or the number of failed edges follows the special distribution, the rankings are coincident with the results generated from some traditional importance measures. When component’s failure sequence follows a saturated nonhomogeneous Poisson process, the proposed importance measures are equivalent to the structural importance measure as time approaches zero or infinite. Finally, numerical examples are provided to demonstr