On the reliability and the optimal replacement policy of a <i>k</i>-out-of-<i>n</i> system in a discrete shock model

In this paper, we consider a [Formula: see text]-out-of-[Formula: see text] system subjected to shocks arriving through a Bernoulli process, that is, the shocks can change from their initial values to only two possible values. Assume that each input shock to the system can lead to the failure of some components. Two scenarios are examined here, in the first scenario it is assumed that at least one component fails with probability one, and in the second scenario we assume that at least one component fails with probability [Formula: see text]. Then, the reliability and an optimal replacement strategy for a [Formula: see text]-out-of-[Formula: see text] system under these scenarios are investigated. With the proposed scenarios, a closed form for the likelihood function for systems in Markovian and non-Markovian regimes are then derived. A model for the preventive replacement policy of the system is also presented. Numerical examples are also given to illustrate the theoretical results.