Stability analysis of an n-unit series repairable system with one repairman doing other work

This paper is devoted to studying the stability and reliability indices of an n-unit series repairable system with a repairman doing other work. By strong continuous semi-group theory, the paper first shows that both the dynamical solution and the steady solution of the system are existence and uniqueness. Then, by analyzing the spectrum distribution of the system operator and by the quasi-compactness of the strong continuous semi-group which generated by the system operator, this paper proves that the dynamical solution converges to the steady solution exponentially. Moreover, the steady solution is just the eigenvector corresponding to eigenvalue 0 of the system operator. In the end of the paper, by normalizing the steady solution, some reliability indices, such as steady availability are presented.