【作者】 艾合买提·阿不来提；

【导师】 艾尼·吾甫尔；

【作者基本信息】 新疆大学 ， 基础数学， 2009， 硕士

【摘要】 本文共分两章。第一章分二节。第一节回顾可靠性理论的历史。第二节中首先介绍补充变量方法,然后提出本文要研究的问题。第二章共分二节。第一节中首先介绍带临界和非临界故障的可修k/N:G冗余表决系统的数学模型,接着引入状态空间,主算子及其定义域,然后将该模型转化成Banach空间中的抽象Cauchy问题,最后介绍其他学者关于此模型的研究成果。第二节中当修复率为常数时讨论此模型的时间依赖解的渐近性质。首先研究此模型主算子的特征值,证明此主算子在左半复平面存在有限多个特征值。其次研究此主算子的共轭算子的特征值,证明该共轭算子在左半复平面也存在有限多个特征值。然后研究此主算子的预解式并给出预解式的表达式。最后运用这些结果与残数定理给出研究该模型过程中出现的投影算子的表达式。由此推出该模型时间依赖解指数收敛于该模型的稳态解。更多还原

【Abstract】 This thesis consists of two chapters. The first chapter is divided into two sections. In section 1, we introduce briefly the history of reliability theory . In section 2, we introduce supplementary variable technique and then put forward the problems we are concerned in this thesis. Chapter 2 consists of two sections. In section 1, firstly we present the mathematical model of the k-out-of-N:G redundant system with repair and multiple critical and non-critical errors, 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 system obtained by other researchers. In section 2, when the repair rates are constants we study asymptotic property of the time-dependent solution of this model. First we study eigenvalues of the operator corresponding to this model and prove this operator exists finite eigenvalues in the left half complex plane. Second we study eigenvalues of the adjoint operator of this operator and also it has finite eigenvalues in the left half complex plane, third we study the resolvent of this operator and give the expression of its 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 this model converges exponentially to the steady-state solution of this model.更多还原