Asymptotic Property Of The Time-Dependent Solution Of The Model Describing A Repairable, Standby Human & Machine System

This thesis consists of two chapters. The first chapter is divided into two sections. In the first section, we introduce briefly the history of the reliability theory and in the second section, we introduce supplementary variable technique and then put forward the problems we are concerned in this paper. The second chapter consists of two sections. In Section 1, firstly we present the mathematical model describing a repairable, standby human & machine system, next we convert the model into an abstract Cauchy problem by introducing state space, operators and their domains, lastly we introduce the main results obtained by our predecessors. In Section 2, we study asymptotic property of the time-dependent solution of the model. We study the spectral properties of the operator corresponding to the model and obtain that zero is an eigenvalue of the operator and its adjoint operator with geometric multiplicity and algebraic multiplicity one, all points on the imaginary axis except for zero belong to the resolvent set of the operator. Thus we derive that the time-dependent solution of the model strongly converges to its steady-state solution as times tends infinite.