Analyse de la fiabilité des systèmes cohérents.

Abstract

The 𝑘-out-of-𝑛 system represents a redundancy structure widely used in reliability engineering. It is a system that fails (or operates) if and only if at least k of its components fail (or operate). This type of system is encountered in fields such as telecommunications, data transmission, redundant networks, etc. In this work, we mainly focus on 𝐾𝑁 -out-of-𝑁 systems, where 𝐾𝑁 = 𝑁 − 𝑗, which are widely studied in optimization problems. It involves a cost optimization study for these systems when the number of components is unknown and is assumed to be a random variable with a predefined probability distribution. The goal is to find the optimal number of components and the optimal replacement time that minimize the defined average costs. In cases where 𝑁 follows one of the distributions : truncated geometric, truncated Poisson, or truncated logarithmic, it has been shown that when the system’s failure distribution follows an exponential or Weibull law, the average cost reaches its minimum for a unique value of 𝜃, the parameter of the considered 𝑁 distribution. That is to say, by substituting the found value into the expression of the expected value 𝐸(𝑁) of the given distribution, we find the optimal number of components. The same method is used to find the optimal replacement time. Using traditional methods, it is shown that for these same distributions, the optimal replacement time exists and is unique. All these results are illustrated by numerical applications. Figures and tables have been created to demonstrate the existence of the optimal number and provide the corresponding cost value, as well as those for the replacement time and its associated cost. Finally, some comparisons between the different studied cases are presented to show the effect of the chosen distributions on the results.