Kn-out-of-n: F System with a Random Number of Components

Abstract

k-out-of-n systems are among the most important models in practical engineering, they have been extensively studied in the past decades. The system fails if and only if there exist at least k failed components. Generally these systems are considered with a fixed number of components. However, many systems have a variable number of components which can be considered a random variable. In the present paper, we studied the [Formula: see text]-out-of-[Formula: see text]: F system, we assume that the system has an unknown number of components, and we search for this number, assuming it is a random variable, by optimizing the system’s cost. This variable is supposed following a power series distribution such as geometric, Poisson and logarithmic distributions. This work discusses the case where [Formula: see text] for optimization problems. Optimal replacement policies, the optimal number of components and the optimal replacement time are determined to make maintenance and avoid failure, this will minimize the mean cost rates. We started with the case of constant number of components, where we give real and approximate values of the MTTF, optimal number of components and optimal replacement time, then, we considered variable number of components, where the results are compared to the constant case. Numerical examples are also given to illustrate our results.