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.

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We investigate Gaver's parallel system attended by a cold standby unit and a repairman with multiple vacations. By using C 0-semigroup theory of linear operators in the functional analysis, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

By using the theory of C0-semigroups and spectral theory of positive operators, we prove well-posedness of the parallel maintenance system with two components and study the asymptotic behavior of the time-dependent solution.

We investigate an N-unit series repairable system with a repairman doing other work. By analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we show that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution.