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Stability analysis of a new kind series system. (English) Zbl 1193.93150
Summary: An $$n$$-unit series repairable system with a repairman doing other work is discussed in this paper. We are devoted to studying the unique existence of the solution and the exponential stability of the system. $$C_0$$-semigroup theory is used to prove the existence of a unique non-negative solution of the system. Then by analysing the spectra distribution of the system operator and the quasi-compactness of the system, we derive that the transient solution of the system exponentially converges to the non-negative steady-state solution, which is the eigenvector corresponding to eigenvalue 0 of the system operator. Some reliability indices of the system and the strategy of the repairman doing other work in his idle time are presented at the end of the paper.

##### MSC:
 93D20 Asymptotic stability in control theory 93B28 Operator-theoretic methods 93C25 Control/observation systems in abstract spaces
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