In the reliability analysis of systems, all system components are often assumed independent and failure of any component does not depend on any other component. One of the reasons for doing so is that considerations of calculation and elegance typically pull in simplicity. But in real-world applications, there are very complex systems with lots of subsystems and a choice of multiple components that may interact with each other. Therefore, components of the system can be affected by the occurrence of a failure in any of the components. The purpose of this paper is to give an explicit formula for the computation of the reliability of a system with two parallel active components and one spare component. It is assumed that parallel components are dependent and operate simultaneously. Two distributions of Freund’s bivariate exponential and Marshall–Olkin bivariate exponential are used to model dependency between components. The results show that the reliability of the system with Freund’s bivariate exponential distribution has lower reliability. The circumstances that lead to them, namely load-sharing in the case of Freund, results in lower reliability. Finally, a numerical example is solved to evaluate the proposed model and sensitivity analysis is performed on the system reliability function. The obtained results show that because the proposed model is influenced by the dependency, compared to traditional models, it has the characteristic of leading to reduced time to (first) failure for achieving specified reliability.
Fuzzification of Linear Combination Based on Marshall-Olkin Bivariate Exponential Distribution