No abstract

We investigate the solution of a repairable parallel system with primary as well as secondary failures. By using the method of functional analysis, especially, the spectral theory of linear operators and the theory ofC0-semigroups, we prove well-posedness of the system and the existence of positive solution of the system. And then we show that the time-dependent solution strongly converges to steady-state solution, thus we obtain the asymptotic stability of the time-dependent solution.

We investigate the solution of an N-unit series system with finite number of vacations. By using -semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

We investigate the Gnedenko system with one repairman who can take vacations. Our main focus is on the time asymptotic behaviour of the system. Using C 0 -semigroup theory for linear operators we first prove the well-posedness of the system and the existence of a unique positive dynamic solution given an initial value. Then 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.

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.