We investigate a series-parallel repairable system consisting of three-unit with multiple vacations of a repairman. By using -semigroup theory of linear operators in the functional analysis, we prove that the system is well-posed and has a unique positive dynamic solution.

We study a series-parallel repairable system consisting of three units with multiple vacations of a repairman. We first show that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of the operator, and then we prove that the semigroup generated by the operator is irreducible. By combining these results with our previous result we deduce that the dynamic solution of the system converges strongly to its steady-state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.