Impact of copulas for modeling bivariate distributions on system reliability

A copula-based method is presented to investigate the impact of copulas for modeling bivariate distributions on system reliability under incomplete probability information. First, the copula theory for modeling bivariate distributions as well as the tail dependence of copulas are briefly introduced. Then, a general parallel system reliability problem is formulated. Thereafter, the system reliability bounds of the parallel systems are generalized in the copula framework. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that the system probability of failure of a parallel system under incomplete probability information cannot be determined uniquely. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases greatly with decreasing component probability of failure. The maximum ratio of the system probabilities of failure for the other copulas to those for the Gaussian copula can happen at an intermediate correlation. The tail dependence of copulas has a significant influence on parallel system reliability. The copula approach provides new insight into the system reliability bounds in a general way. The Gaussian copula, commonly used to describe the dependence structure among variables in practice, produces only one of the many possible solutions of the system reliability and the calculated probability of failure may be severely biased.