Semigroup Method for a System Consisting of a Reliable Machine, an Unreliable Machine and a Storage Buffer with Finite Capacity

We investigate a system consisting of a reliable machine, an unreliable machine and a storage buffer with finite capacity. This system can be described by a group of partial differential equations with integral boundary conditions. By using C-0-semigroup theory of functional analysis, we prove the well-posedness and the existence of a unique positive dynamic solution of the system. And then through analyzing the spectra distribution of the system operator and irreducibility of the semigroup generated by the system operator, we deduce that the dynamic solution converges strongly to steady state solution, thus we obtain the asymptotic stability of the dynamic solution.