Reliability Analysis for Coherent System using Copula

In this paper, we study the estimation error in coherent systems, with a particular focus on $2$-out-of-$3$ configurations. Coherent systems, which include series, parallel, and general $k$-out-of-$n$ systems, form a fundamental class of reliability models where the system lifetime depends on the joint performance of its components. We formulate analytical expressions to measure the impact of component dependence on system lifetime estimation, represented as the difference between dependent and independent scenarios. To capture realistic dependence structures among component lifetimes, we employ copula functions, including Archimedean, Farlie–Gumbel–Morgenstern, and extreme-value copulas, which allow flexible modeling of joint behavior. Moreover, we illustrate our findings with a range of examples and discussions. In addition, we conduct a simulation-based study to further illustrate the application of our results. Our findings provide valuable insights into the impact of dependence on system reliability and offer practical tools for accurate error estimation in coherent systems.