Reliability Measures of a Three-State Complex System: A Copula Approach

Improvement in reliability and production play a very important role in system design. The two key factors, considered in predicting system reliability, are failure distribution of the component and system configuration. This research discusses the mathematical modeling of a highly reliable complex system, which is in three states i.e. normal, partial failed (degraded state) and complete failed state. The system, partial failed is due to the partial failure of internal components or redundancies and completely failed is due to catastrophic failure of the system. Repair rates are general functions of the time spent. All the transition rates are constant except for one transition where two types of repair namely exponential and general possible are tackled with the help of Gumbel-Hougaard family of copula. By employing the supplementary variable technique, Laplace transformation, various transition state probabilities, availability and cost analysis (predictable profit) are obtained along with steady-state behavior of the system. Inversions have also been carried out so as to obtain time dependent probabilities, which find out availability of the system at any instant. At last some special cases of the system have been taken.