Spectral analysis of a two unit deteriorating standby system with repair

In this paper, we analyze the spectra and stability of a system consisting of a working unit and repair unit, in which the working unit consists of one main unit and one standby unit, while the standby unit may deteriorate in its standby mode. Firstly, we formulate the problem into a suitable Banach space. And then we carry out a detailed spectral analysis of the system operator. Based on the spectral analysis and C0-semigroup theory, we prove the existence of positive solution and finite expansion of the solution according to its eigenvectors. As a consequence we get that its dynamic solutions converges exponentially to the steady-state solution. Finally, we derive some reliability indices of the system. Key–Words: C0-semigroup theory, dynamic solution, steady-state, exponential stability, availability.