Analysis of a complex system with common cause failure and two types of repair facilities with different distributions in failure

This paper through a systematic view presents a mathematical model of a complex system that can fail in n-mutually exclusive ways of total failure or due to common cause failure. The system has two modes, namely normal and failed, with two types of repair facility. Repair rates are arbitrary functions of the time spent. All transition rates are assumed to be constant except at one where two types of failure, namely logarithmic and constant are possible. By employing a supplementary variable technique, Laplace transformation and Gumbel-Hougaard family copula, various transition state probabilities, availability and cost analysis (expected profit) are obtained along with steady-state behaviour of the system. Time-dependent probabilities and availability are also obtained by evaluating inversions. Finally, some special cases of the system and their plots are also appended.