Stability Analysis Of A Repairable System With Two Same Units

The analysis and research on system reliability is regarded as a major concern, in which one of the critical issues is repairable system modeling and analysis. It is one whose function can be restored through various means when the components are fail or degrade. Many scholars have tried to model repairable system and study the associated functions. But most of them still stay in the analysis of the wellposedness of the system and the stability of the solution, and fail to go further, leading to application restriction. So the value isn't very good.This paper mainly studies the system model through functional model. First of all, by selecting the appropriate state space, the problem is converted to research on abstract development. With the operator semigroup knowledge, solutions of the systems is proved. As the mathematical model we study is of practical significance, the existence of positive solutions is of great importance. To this end, we study the existence of positive solutions and its conservative nature. We conduct a detailed analysis and research for the stability of the system. By proving that the system exists strictly dominant eigenvalue, we get a further conclusion that the dynamic solution converge in the form of exponential stability to the system's steady-state solution. In order to study the reliability of the system, we study the spectral distribution of our system. and get the conclusion that there is no spectrum points except zero in the imaginary axis.We find that when the time t is larger than a certain moment, the system dynamic solution converges exponentially to its steady-state solution, which is exponentially stable. For a practical system, we certainly hope that t is a short time. In this paper, we do numerical simulation in the case when the nature repair rateμc(x) of the system is the piecewise function, and get the corresponding estimation on t. Finally, we study that the three groups parameters impact on system performance respectively, and provide some reference for the choice of maintenance strategy, which is a new step for the stability.