Reliability analysis of a simple repairable system

We consider a new kind of simple repairable system consisting of a repairman with multiple delayed-vacation strategy. A common technique in reliability studies is to substitute the steady-state reliability indexes for instantaneous ones because the dynamic solution of the system is difficult or even impossible to obtain. However, this substitution is not always valid. Therefore, it is important to study the existence, uniqueness and expression for the system's dynamic solution, and to discuss the system's stability. The purpose of this paper is threefold: to study the uniqueness and existence of the dynamic solution, and its expression, using C0-semigroup theory; to discuss the exponential stability of the system by analysing the spectral distribution and quasi-compactness of the system operator; to derive some reliability indexes of the system from an eigenfunction point of view, which is different from the traditional Laplace transform technique, and present a profit analysis to determine the optimal vacation time in order to achieve the maximum system profit.