Exact Reliability Formula for Consecutive- $k$ -Out-of- $n$ Repairable Systems

In this paper, consecutive-k -out-of-n repairable systems are studied. We assume that both the working time, and the repair time of each component are exponentially distributed, and every component after repair is as good as new. Each component is either a key component, or an ordinary component so we can adopt a priority repair rule for key components. First, the state transition probabilities as well as their Laplace transforms for the systems are derived. Second, several fundamental reliability indices (including availability, the mean time to the first failure, reliability, and the rate of occurrence of failure) of the systems are obtained explicitly. Finally, one example is shown to explain the model, and the methodology developed in this paper.