Operator semi-group of density evolution equation for a repairable redundant system with two same components

Abstract

For a repairable redundant system consisting of two same components with exponential lifetime and general repair time distribution, the probability densities of the system in some state at time t were determined by a group of ordinary and partial differential equations, called density evolution equations. It was proved that the time-dependent solution of the density evolution equations uniquely exists and strongly converges to its steady state density solution by a semi-group method. In this proof, it is not necessary to suppose that the repair rate function is bounded. The technique of the proof is valuable for many density evolution equations.