Analyzing component failures in series-parallel systems with dependent components

This paper investigates a series-parallel system comprising $N$ independent subsystems with interchangeable dependent components, a prevalent reliability structure in engineering and network design. The primary aim of this research is to derive the joint probability distribution of the number of failed components within these configurations, considering component dependence and varying distributions across subsystems. This approach reflects a more realistic scenario than previously explored in the literature. Initially, the analysis is conducted for systems with two subsystems and subsequently extended to encompass configurations with $N$ subsystems. The study also evaluates key reliability metrics including the average number of failed components and the mean time to failure (MTTF) of the entire system, theoretically proving that the system’s MTTF increases with the number of components under certain sufficient conditions. In addition to probabilistic analysis, an optimization problem is addressed to determine the optimal allocation of components within each subsystem. The objective is to minimize the average cost associated with corrective maintenance, thereby enhancing the cost-effectiveness of system operation.