Study Of A Cold Standby Parallel System Consisting Of Two Repairable Units With Two States

This thesis is divided into two chapters, the first chapter is split two sections. In Section 1, we introduce creation, development and current research situation of reliability theory. In Section 2, we firstly introduce creation and thought of supplementary variable technique, then put forward the problem that we will study in this thesis. The second chapter is divided into three sections. In Section 1, firstly we introduce the mathematical model of a cold stanby parallel system consisting of two repairable units with two states, then we convert the model into an abstract Cauchy problem by introducing state space, operators and their domain. In Section 2, we prove the existence and uniqueness of the time-dependent solution of the model by using the Hille-Yosida theory and the Phillips theorem. In Section 3, we discuss asymptotic property of the solution. Through direct estimating the norm of resolvent of the operator corresponding to the model we obtain that all points on the imaginary axis belong to the resolvent set of the operator. Thus, by combining this result and the result in Section 2 with the ABLV theorem we deduce that the time-dependent solution of the model strong asymptotic stable, that is, the time-dependent solution of the model converges to zero by means of the norm in state space.