Distribution of failures of a coherent system under the condition of operating the system with discrete lifetime

In this article, the number of failures of a coherent system has been studied under the assumption that the lifetime of system components are non-distributed discrete and dependent random variables. First, the probability that exactly i Failure i=0, ..., n-k, in a system $k$ From n Under the condition that the system at the time of monitoring t it works it will be counted. In the following, this result has been generalized to other coherent systems. In addition, it has been shown that in the case of independence and co-distribution of component lifetimes, the probability obtained is consistent with the corresponding probability in the continuous state obtained in the existing literature. Finally, by presenting practical examples, the behavior of this probability has been investigated in the case that the system components have interchangeable and necessarily non-distributed lifetimes