带临界和非临界故障的可修k/N：G冗余表决系统研究中出现的投影算子的表达式及其应用
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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。
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.
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.
引文
[1]Who Kee Chung.Stochastic analysis of k-out-of-N:G redundant system with repair and multiple critical and non-critical errors.Microelectronics and Reliability[J].1995,35:1429-1431.
[2]Xu Hou-bao,Guo Wei-hua,Yu Jing-yuan,Zhu Guang-tian.Asymptotic stability of k-out-of-N:G redundant system.Acta Analysis Functionalis Applicata[J].2004,6:209-219.
[3]Geni Gupur,Xue-zhi Li,Guang-tian Zhu.Functional Analysis Method in Queueing Theory[M].Herdfortshire:Research Information Ltd.2001.
[4]热娜·艾合买提.带临界和非临界故障的可修k/N:G冗余表决系统的进一步研究[D].乌鲁木齐:新疆大学硕士学位论文,2008.
[5]曹晋华,程侃.可靠性数学引论[M].北京:科学出版社,1986.
[6]W.Weibull.A statistical theory of the strength of materials.Ing.Vetenskaps Akad.Handl[J].1939,151:1-45.
[7]D.R.Cox.The analysis of non-markovian stochastic Processes by the inclusion of supplementary variables.Proc.Camb.Phil.Soc.[J].1955,51:433-441.
[8]D.P.Gaver.Time to failure and availibility of parallel redundant systems with repair.IEEE Transactions on Reliability[J].1963,12:30-38.
[9]D.G.Linton,Some advancements the analysis of two-unit parallel redundant systems .Microelectronics & Reliability[J].1976,15:39-46.
[10]K.Adachi,M.Kodama,M.Ohashi.k-out-of-n:G system with simultaneous failure and three repair policies.Ibid[J].1979,19:351-361.
[11]R.Subramanian,N.Ravichandran.Probabilistic analysis of a two-unit parallel redundant system.Microelectronics & Reliability[J].1979,19:321-323.
[12]M.Ohashi等.Microelectronics & Reliability[J].1980,20:471-476.
[13]史定华.两部件并行系统可靠性分析的某些新推进.应用数学学报[J].1985,8(2):159-172.
[14]艾尼·吾甫尔.一类两个相同部件并联的可修系统的适定性.应用泛函分析学报[J].2001,3:188-192.
[15]Geni Gupur,Li Xue-zhi.Semigroup method for a mathematical model in reliability analysis.Journal of Systems Science and Systems Engineering[J].2001,10:137-147.
[16]艾尼·吾甫尔,李学志.一个可靠机器,一个不可靠机器和一个缓冲库构成的系统分析.系统工程理论与实践[J].2002,22(2):29-36.
[17]Geni Gupur.Well-posedness of a reliability model.Acta Analysis Functionalis Applicata [J].2003,5:193-209.
[18]艾尼·吾甫尔,郭宝珠.一类可靠性模型解的渐近稳定性.应用泛函分析学报[J].2002,4:252-261.
[19]Geni Gupur.Well-posedness of describing a repairable standby,human & machine system.Journal of Systems Science and Complexity[J].2003,16:483-493.
[20]艾尼·吾甫尔,依里哈木·玉素甫,尼亚孜·苏来曼.一类两个相同部件并联的可修系统解的渐近性质.新疆第五届青年学术年会论文集[C].乌鲁木齐:新疆人民出版社,2004,646-649.
[21]Abdukerim Haji,Geni Gupur.Asymptotic property of the solution of a reliability model.International Journal of Mathematical Sciences[J].2004,3(1):161-195.
[22]Geni Gupur,Guo Bao-zhu.Asymptotic property of the time-dependent solution of a reliability model.Journal of Systems Science and Complexity[J].2005,18(3):319-339.
[23]Geni Gupur.Asymptotic stability of the time-depedent solution of a reliability model .Acta Analysis Functionalis Applicata[J].2005,7:299-316.
[24]Geni Gupur.Asymptotic property of the solution of a repairable,standby,human and machine system.International Journal of Pure and Applied Mathematics[J].2006,28:35-54.
[25]周俊强,艾尼·吾甫尔.一类可靠性模型研究中出现的投影算子的表达式及其应用.新疆大学学报(自然科学版)[J].2007,24:379-389.
[26]朱海娟,艾尼·吾甫尔.一类可靠性模型的另一个特征值.应用泛函分析学报[J].2008,10:175-186.
[27] Roger A.Horn, Charles R.Johnson. Matrix Analysis [M]. London: Cambridge University Press, 1985.
[2]Xu Hou-bao,Guo Wei-hua,Yu Jing-yuan,Zhu Guang-tian.Asymptotic stability of k-out-of-N:G redundant system.Acta Analysis Functionalis Applicata[J].2004,6:209-219.
[3]Geni Gupur,Xue-zhi Li,Guang-tian Zhu.Functional Analysis Method in Queueing Theory[M].Herdfortshire:Research Information Ltd.2001.
[4]热娜·艾合买提.带临界和非临界故障的可修k/N:G冗余表决系统的进一步研究[D].乌鲁木齐:新疆大学硕士学位论文,2008.
[5]曹晋华,程侃.可靠性数学引论[M].北京:科学出版社,1986.
[6]W.Weibull.A statistical theory of the strength of materials.Ing.Vetenskaps Akad.Handl[J].1939,151:1-45.
[7]D.R.Cox.The analysis of non-markovian stochastic Processes by the inclusion of supplementary variables.Proc.Camb.Phil.Soc.[J].1955,51:433-441.
[8]D.P.Gaver.Time to failure and availibility of parallel redundant systems with repair.IEEE Transactions on Reliability[J].1963,12:30-38.
[9]D.G.Linton,Some advancements the analysis of two-unit parallel redundant systems .Microelectronics & Reliability[J].1976,15:39-46.
[10]K.Adachi,M.Kodama,M.Ohashi.k-out-of-n:G system with simultaneous failure and three repair policies.Ibid[J].1979,19:351-361.
[11]R.Subramanian,N.Ravichandran.Probabilistic analysis of a two-unit parallel redundant system.Microelectronics & Reliability[J].1979,19:321-323.
[12]M.Ohashi等.Microelectronics & Reliability[J].1980,20:471-476.
[13]史定华.两部件并行系统可靠性分析的某些新推进.应用数学学报[J].1985,8(2):159-172.
[14]艾尼·吾甫尔.一类两个相同部件并联的可修系统的适定性.应用泛函分析学报[J].2001,3:188-192.
[15]Geni Gupur,Li Xue-zhi.Semigroup method for a mathematical model in reliability analysis.Journal of Systems Science and Systems Engineering[J].2001,10:137-147.
[16]艾尼·吾甫尔,李学志.一个可靠机器,一个不可靠机器和一个缓冲库构成的系统分析.系统工程理论与实践[J].2002,22(2):29-36.
[17]Geni Gupur.Well-posedness of a reliability model.Acta Analysis Functionalis Applicata [J].2003,5:193-209.
[18]艾尼·吾甫尔,郭宝珠.一类可靠性模型解的渐近稳定性.应用泛函分析学报[J].2002,4:252-261.
[19]Geni Gupur.Well-posedness of describing a repairable standby,human & machine system.Journal of Systems Science and Complexity[J].2003,16:483-493.
[20]艾尼·吾甫尔,依里哈木·玉素甫,尼亚孜·苏来曼.一类两个相同部件并联的可修系统解的渐近性质.新疆第五届青年学术年会论文集[C].乌鲁木齐:新疆人民出版社,2004,646-649.
[21]Abdukerim Haji,Geni Gupur.Asymptotic property of the solution of a reliability model.International Journal of Mathematical Sciences[J].2004,3(1):161-195.
[22]Geni Gupur,Guo Bao-zhu.Asymptotic property of the time-dependent solution of a reliability model.Journal of Systems Science and Complexity[J].2005,18(3):319-339.
[23]Geni Gupur.Asymptotic stability of the time-depedent solution of a reliability model .Acta Analysis Functionalis Applicata[J].2005,7:299-316.
[24]Geni Gupur.Asymptotic property of the solution of a repairable,standby,human and machine system.International Journal of Pure and Applied Mathematics[J].2006,28:35-54.
[25]周俊强,艾尼·吾甫尔.一类可靠性模型研究中出现的投影算子的表达式及其应用.新疆大学学报(自然科学版)[J].2007,24:379-389.
[26]朱海娟,艾尼·吾甫尔.一类可靠性模型的另一个特征值.应用泛函分析学报[J].2008,10:175-186.
[27] Roger A.Horn, Charles R.Johnson. Matrix Analysis [M]. London: Cambridge University Press, 1985.