On the status of component failures in a working coherent system when the lifetimes are DNID random variables

In this paper, we investigate the number of failed components in a coherent system with possibly dependent and not necessarily identically distributed component lifetimes. Based on two different partial information about the status of component failures, obtained either with a single inspection or under double monitoring of the system, we compute the probability of the given number of failures in a working coherent system. In the computations, we use the concept of minimal path sets of the system. These results enable us to establish explicit expressions for survival functions of two types of residual lifetimes of a used system. For illustration, we provide numerical examples of coherent systems composed of discretely or continuously distributed component lifetimes.