Optimal inspection interval for a k-out-of-n system with non-identical components

In this paper, we develop a continuous-time discrete-state model for periodic inspection of a k-out-of-n cold-standby system with non-identical components. A perfect switching system detects the components' failures, and the failed component(s) are repaired during the next inspection interval, and then added to the standby queue. The system can be in different states depending on the combination of working components and the order of the components on the standby queue at the beginning of an inspection interval. We present a matrix-based approach to determine the system states and calculate the system-states transition probabilities and the transition matrix. We calculate the expected total cost of the inspection intervals by determining the system state at the beginning and end of each inspection interval and calculating the inspection cost matrix. The expected total inspection cost consists of the system downtime cost, components repair cost, system repair cost, and system inspection costs. Finally, we minimize the system's expected total cost by determining the system's optimal inspection interval. The results show that determining the optimal inspection interval decreases the system's total inspection interval cost up to 60 % in comparison with the cases when the inspection interval is selected arbitrarily.