A repairable system with warning device and repairman vacation

This paper considers a simple repairable system with a warning device which can send an alarm when the system is not in good condition and a repairman who can have delayed-multiple vacations and carry out overhaul or minor repair for the system according to the defects. By using Markov renewal process theory and the probability analysis method, the system is described into a group of integro-differential equations. Then the unique existence and stability of the system dynamic solution are presented by using the strongly continuous semigroup theory. On the basis, some reliability indexes and applications (such as the comparisons of indexes and profit of systems with and without warning device), as well as numerical analysis are discussed at the end of the paper.