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a b s t r a c tIn this paper, we deal with a new model of an n-unit series repairable system, in which a concept of a repairman with multiple-delayed vacation is introduced and the impact on the system reliability due to a replaceable facility is also considered. This paper is devoted to studying the unique existence and stability of the system solution. C 0 -semigroup theory is used to prove the existence of a unique nonnegative time-dependent solution of the system. Then by analyzing the spectra distribution of the system operator, we prove that the dynamic solution of the system asymptotically converges to the nonnegative steady-state solution which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Furthermore, we discuss the exponential stability of the system in a special case. Some reliability indices of the system are also studied and the optimal vacation time is analyzed at the end of the paper.

We develop a mathematical model for the disease which can be transmitted via vector and through blood transfusion in host population. The host population is structured by the chronological age. We assume that the instantaneous death and infection rates depend on the age. Applying semigroup theory and so forth, we investigate the existence of equilibria. We also discuss local stability of steady states.

The exponential stability of a multi-state device is discussed in this paper. We present that the C0-semigroup generated by the system operator is quasi-compact and irreducible. It is known that 0 is a simple eigenvalue of the system operator. In combination with this, we obtain that the time-dependent solution exponentially converges to the steady-state solution, which is the positive eigenfuction corresponding to the simple eigenvalue 0.