Discrete time three-state k-out-of-n system’s failure and numbers of components in each state

In this paper we consider three-state $k$-out-of-$n$ system composed of components which lifetimes are modeled by independent and identically distributed discrete random variables. The primary focus is the random vector representing the numbers of components in each state. We derive its joint distribution. For illustration, we provide examples of the systems with components with geometrically distributed lifetimes following the Markov degradation process.