Probabilistic model–based multi-integration formulas for quantifying a generalized minimal cut sequence

Many sequence- and function-dependent failure behaviors may exist in fault-tolerant systems, in which the failure of the systems depends not only on the combination of basic events but also on their failure sequences. Such systems can be modeled by dynamic fault tree with Priority AND gate, Spare gate, Sequence Enforcing gate and Functional Dependence gate. The existing analytical methods for analyzing these dynamic gates are mainly Markov-based or inclusion-exclusion-based approaches. Compared with Markov approaches, the inclusion-exclusion-based method presents notable advantages. However, to the author's knowledge, no references by far have presented general multi-integration formulas applicable for solving a generalized minimal cut sequence, which is an unavoidable problem in inclusion-exclusion-based method. In this article, new multi-integration formulas are proposed based on traditional probabilistic models, which are applicable for solving a generalized minimal cut sequence as well as any dynamic gate. The rigorous derivation processes are given. The application and advantages of our proposed approach are demonstrated through a case study and the results indicate our proposed approach is reasonable.