On the number of failed components in a coherent operating system

In this paper, we investigate the number of failed components in a coherent system. We consider an (n−m+1)(n−m+1)-out-of-nn system and we compute the probability that there are exactly ii failures, i=0,…,m−1i=0,…,m−1, in the system under the condition that it is operating at time tt. Several properties of the proposed time dependent probabilities are studied. The results are then extended to coherent systems consisting of nn components with an arbitrary signature vector.