Analysis Of A N-Unit Series Repairbale System With Repairman Single Delay Vacation

Currently, reliability study is an important area of research. Reparable system is one of the most important systems discussed in the reliability theory. Many scholars, at home and abroad, have done lots of work about the system. However, all studies are based on two hypotheses:(1)the system concerned has a unique nonnegative time-dependent solution;(2) the solution of the system is asymptotic stability. However, whether system is exponential stability that is still unsolved.The N-unit series repairable system with the repairman single delay vacation is studied in the paper. Firstly, the system model of interest is established and the corresponding parameters and conditions are given. It is translated into an abstract Cauchy problem in a Banach space. Secondly, by anaryzing the closeness, denseness and dissipative property of the system operator. We obtain a positive compression C0-Semigroup of the system operator and the unique existence of the nonnegative time-dependent weak solution of the system is proved. Thirdly, in special cases, the existence and uniqueness of the nonnegative time-dependent strong solution of the system is obtained by use of the Volterra operator equation in a Banach space. Moreover, by analyzing the spectral distribution of the system operator and its dual operator, we prove that zero is the simple eigenvalue of the dual operator and the asymptotic stability of the system is given. And, the paper proves to be quasi-compactness of the system operator. Hence, we obtain that the nonnegative time-dependent solution of the system converges to the steady solution by the way of exponential. At last, using the theory of Banach space, the optimal control problem of steady-state solution of is studied, and the existence and uniqueness for the unit of optimal are proved. The results show that the steady-state availability of the system can reach its expected value in the limited time.