Instantaneous reliability index of a typical repairable system

A two-state system was taken as an example to show the method that we derived for analyzing the instantaneous reliability index, which was an important issue in the field of reliability. The paper first described the system as an abstract Cauchy problem by choosing suitable operators and state spaces. Then, with the Trotter-Kato theorem and strong continuous semi-group theorem, the method of solving the instantaneous reliability index of the repairable system was constructed. The convergence of the method was also proved in theory in this paper. To show the effectiveness of this method, some numerical examples were given at the end of the paper.