Spectral Analysis Of A Repairable System With Standby Unites

In this paper, we introduce a repairable system with standby unites, which consists of two subsystems. One of the subsystems is composed of a main unit and a standby unit, while the other one includes only one unite. The system may experience two kinds of failure:the hardware failure (human error) and the natural failure. Through the discussion of well-posedness, positive solutions and spectral distribution properties of the system with the help of the operator semigroup theory and functional analysis methods, we tend to indicate the uniqueness, conservation and the essential spectrum distribution of the system's solution.When studying the well-posedness of the system, we normalize the differential inte-gral equations and consider it as Cauchy equation in Banach space to prove the operator is densely defined closed dissipative. However, the presence of feedback item greatly increases the complexity of the problem, therefore we first take out of the feedback item to prove that the operator without the feedback could generate a Co compression semigroup, the we can get the expected results by using the semigroup perturbation theorem. After this, we prove that the system opertor is the dispersion operator, the system-generated semigroup is a positive one, and also verify that the system is con-servative.Next we do the spectrum distribution analysis to discuss the zero points of the eigenvalue equation. After complicated calculation, a coefficient polynomial of the sys-tem is obtained and 0 is a eigenvalue, which means the system has a steady solution. According to perturbation theory, a compact perturbation does not change the operator of the mass. After all these, we make a study on the spectrum distribution of the system opertor. At last, we give the failure rate and repair functions some certain values to make a few numerical simulation maps, and come to the conclution that our analysis of the spectral distribution is correct.