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

This thesis is divided into two chapters. Chapter 1 is divided into two sections. In Section 1, we introduce the generation, development and present research situation of reliability theory. In Section 2, we first introduce the generation and thought of supplementary variable technique, then we put forward the problem that we will study in this thesis. Chapter 2 is divided into three Sections. In Section 1, we first introduce the mathematical model of the cold standby parallel repairable system consisting of two identical units with three states, then by defining the state space, operator and its domain, we convert the model into an abstract Cauchy problem. In Section 2, by using the Hille-Yosida theory and the Phillips theory, we prove that the model has a unique positive time-dependent solution. In Section 3, we discuss the asymptotic property of the time-dependent solution of the model. Through studying the regular points of the adjoint operator of the operator corresponding to the model we obtain the regular points of the operator, then by studying spectrum of the operator we obtain the spectral properties of the operator : all points on the imaginary axis except for zero belong to the resolvent set of the operator, zero is continuous spectrum of the operator. By combining this result and the result of Section 2 with ABLV theorem, we conclude that the time-dependent solution of the model is strongly asymptotic stable, namely, the time-dependent solution of the model converges to zero by the norm.