Mean time to failure of a k-out-of-n system with a single cold standby unit having dependent components

In this article, we consider a k-out-of-n system which is equipped with a single cold standby component. We assume that the components within the k-out-of-n system are exchangeable. The standby component becomes active when the k-out-of-n system experiences a failure. The active cold standby component reboots the system in collaboration with the remaining original active components. The dependency between the residual lifetimes of the survival components and the cold standby component is assumed to be modeled by a copula function C*, while the dependency between the original components of the k-out-of-n system is assumed to be modeled by a symmetric copula function C. Initially, we derive the common cumulative distribution function of the survival components of the k-out-of-n system at the time of the failure of the k-out-of-n system. After that, we compute the mean time to failure of the considered system. We analyze how various dependence parameters among the components impact the system’s reliability.