The Expression Of The Project Operator Appeared During Research Of A Reliability Model And Its Application

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.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 of the system consisting of a reliable machine,an unreliable machine and a storage buffer,next we convert the model into an abstract Cauchy problem by introducing state space,operators and their domains,lastly we introduce the main results about this model obtained by other researchers.In Section 2,first we study eigenvalues of the operator corresponding to this model and prove that±(?)-λη₁-μη₂ are eigenvalues of this operator and there are no other eigenvalues of this operator expect for±(?)-λη₁-μη₂ in the left half complex plane.Second we study eigenvalues of the adjoint operator of this operator and obtain that there are no other eigenvalues of this adjoint operator expect for±(?)-λη₁-μη₂ in the left half complex plane.Third we study the resolvent of this operator and give the expression of this resolvent.Last we give expression of the project operator appearing during research of this model by using the above results and the residue theorem.Thus we deduce that the time-dependent solution of the model converges exponentially to the steady-state solution of the model.