复杂系统平均寿命的估计及性质
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摘要
可靠性数学理论起源于20世纪30年代,最早被应用的领域是机械证明,维修问题。另一个重要应用是将更新问题应用于更换问题。在30年代,威布尔,龚贝尔和爱泼斯坦等人研究了材料的疲劳寿命以及相关的极值问题。在二次世界大战以后,可靠性问题得到了更大的重视,随着在军事技术装备上的应用日新月异,可靠性问题也更多的应用到生活中随处可见的复杂系统问题上。从50年代至今,可靠性这门新兴的学科以惊人的速度发展着,在各个领域都积累了丰富的经验,可靠性理论的应用已经从军事技术领域扩展到了国民经济等许多领域。
系统的可靠度,平均寿命,子系统的重要度等都是评估系统可靠性的重要指标,在航天航空等领域,相对于其他指标,平均寿命有其独特的意义。目前,系统的平均寿命在日常生活中的应用也越来越被重视。
本文主要研究以下几个问题:
首先,了解复杂系统的基本知识要点,描述复杂系统的方法,介绍了最小路径矩阵和最小割集矩阵的定义以及可靠度的解析表达式等相关理论知识。
其次,对子系统寿命变量分布服从指数分布的复杂系统,利用可靠度的解析表达式,给出了系统平均寿命的表达形式。进一步分别在完全数据和截尾数据两种情况下,给出了其估计,并在完全数据形式下,证明了其估计的渐近正态性。
最后,对于子系统寿命变量服从一般分布的情况下,给出了系统平均寿命的估计,并对估计性质进行了研究。
The theory of mathematical reliability dated from 1930’s which had been used in the area of mechanical proof and maintenance at the very beginning. Another important application of this theory was to apply the question of updating to the question of replacement. In 1930’s, the pioneers, Weibull, Gong Bell and Epstein, had done the research about the fatigue life of materials and the relative problem of extremum. After the World WarⅡ, the question of reliability had been focused more, and this question was used to solve the systematic problems in daily life as the improvement of military technology. since 1950’s, the subject of reliability has been developing rapidly and getting rich experiences in different areas. The application of reliability theory is now extends from military technology to all areas of national economy.
The reliability, mean life, importance of the subsystem are the essential indicators of system reliability evaluating, but in the region of aerospace we focus on mean life more than any other indicators. Now, the life expectancy of the system has been taken more serious in everyday life.
Several questions had been investigated as follows:
Firstly, intrducting the basic knowledge of the complex system and the methods for describing complex system, which also introduce the conception of minimum path matrix and minimal cut set matrix as well as the relative theory about the analytical expression of reliability.
Secondly, for the complex system whose every subsystem life having exponential distribution, the mean life of it is given by using function of complex system reliability. Estimation of the system mean life are presented under complete datum as well as censored datum. Furthermore, we prove the asymptotic normality of complex system reliability under complete datum condition.
At last, for the complex system whose every subsystem life having general distribution, the estimation of the complex system is given and the properties are studied.
系统的可靠度,平均寿命,子系统的重要度等都是评估系统可靠性的重要指标,在航天航空等领域,相对于其他指标,平均寿命有其独特的意义。目前,系统的平均寿命在日常生活中的应用也越来越被重视。
本文主要研究以下几个问题:
首先,了解复杂系统的基本知识要点,描述复杂系统的方法,介绍了最小路径矩阵和最小割集矩阵的定义以及可靠度的解析表达式等相关理论知识。
其次,对子系统寿命变量分布服从指数分布的复杂系统,利用可靠度的解析表达式,给出了系统平均寿命的表达形式。进一步分别在完全数据和截尾数据两种情况下,给出了其估计,并在完全数据形式下,证明了其估计的渐近正态性。
最后,对于子系统寿命变量服从一般分布的情况下,给出了系统平均寿命的估计,并对估计性质进行了研究。
The theory of mathematical reliability dated from 1930’s which had been used in the area of mechanical proof and maintenance at the very beginning. Another important application of this theory was to apply the question of updating to the question of replacement. In 1930’s, the pioneers, Weibull, Gong Bell and Epstein, had done the research about the fatigue life of materials and the relative problem of extremum. After the World WarⅡ, the question of reliability had been focused more, and this question was used to solve the systematic problems in daily life as the improvement of military technology. since 1950’s, the subject of reliability has been developing rapidly and getting rich experiences in different areas. The application of reliability theory is now extends from military technology to all areas of national economy.
The reliability, mean life, importance of the subsystem are the essential indicators of system reliability evaluating, but in the region of aerospace we focus on mean life more than any other indicators. Now, the life expectancy of the system has been taken more serious in everyday life.
Several questions had been investigated as follows:
Firstly, intrducting the basic knowledge of the complex system and the methods for describing complex system, which also introduce the conception of minimum path matrix and minimal cut set matrix as well as the relative theory about the analytical expression of reliability.
Secondly, for the complex system whose every subsystem life having exponential distribution, the mean life of it is given by using function of complex system reliability. Estimation of the system mean life are presented under complete datum as well as censored datum. Furthermore, we prove the asymptotic normality of complex system reliability under complete datum condition.
At last, for the complex system whose every subsystem life having general distribution, the estimation of the complex system is given and the properties are studied.
引文
[1] FABRICE G, BERNARD D, EMMANUEL U. Reliability Estimation by Bayesian Method: Definition of Prior Distribution Using Dependability Study[J]. Reliability Engineering and System Safety, 2003, 82(3): 299-306.
[2] SPOTT D A. Normal Likehoods and Relation to Large Sample Theory of Estimation[J]. Biometrika, 1973, 60(2): 457-465.
[3] BARTH D J. The Sampling Distribution of an Estimate Arising in Life Testing[J]. Technometrics, 1963, 3(5): 361-374.
[4] BARLOW E, PROSCHAN F. Mathematical Theory of Reliability[C]. New York Wiley, 1965,320-322.
[5] GUO S X. Efficient method for computing the fault probability of large-scale network system [J]. Systems Engineering and Electronics, 2005, 25(4): 744-747.
[6] LALIT G, Yan O. Radial Distribution System Reliability Worth Evaluation Utilizing the Monte Carlo Simulation Technique[J]. Computers Electrical Engineering, 2001, 27: 273-285.
[7]程述汉,董厚奎.利用次序统计量估计种群的平均寿命[J].生物数学学报, 2000, 15(3): 120-126.
[8] BRYANT R E. Graph-based Algorithm for Boolean Function Manipulation[J]. IEEE Transaction on Comput, 1986, C-35: 677-691.
[9]陈景鹏,金星.不同指数单元的串联系统的平均寿命评定[C].第六届全国计算机应用联合学术会议论文集, 2002: 40-42.
[10]于丹.复杂系统可靠性分析中的若干统计问题与进展[J].系统科学与数学, 2007, 27(1): 68-81.
[11]金星.大型复杂系统平均寿命评定的Monte Carlo方法[J].系统仿真学报, 2005, 17(1): 66-68.
[12]张艳等.基于置信分布的系统可靠度评估蒙特卡罗方法[J].北京航空航天大学学报, 2006, 32(10): 1023-1025.
[13]刘颖.模糊随机不可修复系统的可靠度和平均寿命分析[D].天津:天津大学, 2006: 7-14.
[14] REPICI M, SORNIOTTI A. A General Approach to Study the Reliability ofComplex Systems[C]. Acta Polytechica, 2003, 43(3): 34-40.
[15]李学京,韩筱爽.复杂系统平均寿命综合评估方法研究[J].数学的实践与认识, 2007, 37(2): 110-113.
[16]曹晋华,程侃.可靠性数学引论[M].修订版.北京:高等教育出版社, 2006: 1-12.
[17]陈常顺.平均剩余寿命置信下限的评估[J].中北大学学报, 2007, 28(3): 208-211.
[18]李学京.可靠性综合验证方法研究[J].中国科学院文献情报中心, 2007, 3: 42-47.
[19]陈家鼎.生存分析与可靠性[M].北京:北京大学出版社, 2005.
[20] JANE C, YUAN J. A Sum of Disjoint Products Algorithm Network-reliability Evaluation of Flow Networks[J]. Reliability Engineering and System Safety, 1993, 131: 664-675.
[21]杨军,赵宇.复杂系统平均剩余寿命综合评估方法[J].航空学报, 2007, 28(6): 1351-1354.
[22]具光花,周艳魏.完全样本下威布尔分布平均寿命的评估[J].延边大学学报, 2008, 34(4): 246-249.
[23]周源泉,翁朝曦.可靠性评定[M].北京:科学技术出版社, 1990: 72-79.
[24]张国志.复杂系统可靠性分析[D].北京:北京工业大学, 2009:22-24.
[25] ALBERT H, SNEAD R C. A Comparison of Monte Carlo Technique for Obtaining System Reliability Confidence Limits From Component Test Data[J]. IEEE Transaction on Reliability, 2007, 1(20): 320-332.
[26] LEV V, UTKIN. Interval Reliability of Typical Systems with Partially Known Probabilities[R]. European Journal of Operational Research, 2004, 153(3): 790-802.
[27] FOTUHI F M, SENIOR M A. Novel Approach to Determine Minimal Tie-sets of Complex Network[J]. IEEE Transaction on Reliability, 2004, 53(1): 61-70.
[28]翟伟丽,茆诗松.定时截尾场合下双参数指数分布的参数估计[J].应用概率统计, 2002, 18(2): 197-200.
[29]高峰,温书.定时截尾样本下指数分布平均寿命的假设检验[J].淮阴师范学院学报, 2004, 3(1): 14-15.
[30]陈家鼎,王宏.关于定时截尾寿命试验方案下指数分布情形平均寿命的置信限[R].北京:北京大学, 1995.
[31]王宏.混合截尾寿命试验情形下指数分布平均寿命的置信限[J].应用概率统计, 1996, 12(4): 20-22.
[32] YEY W. The Extension of Universal Generating Function Method to Search for All One-to-many D-minimal Paths of Acyclic Multi-state-arc for Conserv-ation Networks[J]. IEEE Transactions on Reliability, 2008, 57(1): 94-102.
[33] WEI C. A Simple Heuristic Algorithm for Generating All Minimal Paths[J]. IEEE Transactions on Reliability, 2007, 56(3): 488-494.
[34] TANG J. Mechanical System Reliability Analysis Using a Combination of Graph Theory and Boolean Function[J]. Reliability Engineering and System Safety, 2001, 72 (1): 21-30.
[35]范金成.统计推断号引[M].北京:科学出版社, 2001.
[36] SINGH B. Inferences Concerning Exponential Distributions in the Presence of Randomly Right Censored Data with Missing Censored Values, Lifetime Data Analysis[J]. IEEE Transaction on Reliability, 2002, 12(8): 69-88.
[37] CHANG Y. OBDD-Based Evaluation of Reliability and Importance Measures for Multisate Systems Subject to Imperfect Fault Conerage[J]. IEEE transactions on dependable and secure computing, 2005, 2(4): 336-347.
[38]田庭,刘次华.定时截尾缺失数据下指数分布的统计推断[J].华侨大学学报, 2006, 27(1): 20-22.
[39]陈家鼎,房祥忠.任意序贯样本下指数分布均值置信限[J].中国科学(A辑), 2005, 35(4): 1-8.
[40]李林.多次混合截尾寿命试验指数分布平均寿命的统计推断[D].北京:北京大学, 2005: 12-26.
[41]闫霞,于丹,李国英.一类可修威布尔型设各可用度Fiducial推断[J].系统利学与数学, 2004, 24(1): 17-27.
[42]严广松,侯紫燕.不相同定时截尾试验中平均寿命的置信限[J].郑州纺织工程院学报, 1998, 9(3): 19-21.
[43]尹逊汝.Ⅰ型截尾模型修正后平均寿命的非参数估计[J].泰山学院学报, 2007, 29(3): 20-23.
[44] TANG J. Mechanical System Reliability Analysis Using a Combination of Graph Theory and Boolean Function[J]. Reliability Engineering and System Safety, 2001, 72 (1): 21-30.
[45]廖林生,邓文丽.区间数据样本下指数分布平均寿命的矩估计[J].江西科学, 2007, 25(5): 540-545.
[46]张民悦.复杂可修系统的一类可靠性数学基础[J].兰州理工大学学报, 2002, 28(3): 118-120.
[47]荆广珠,房祥忠.非等定时截尾寿命试验方案指数分布情形平均寿命的置信限[J].数理统计与管理, 2000, 04: 46-49.
[48]张定海,张悦民. n元件复杂系统的可靠性分析[J].甘肃科学学报, 2007, 19(2), 131-132.
[2] SPOTT D A. Normal Likehoods and Relation to Large Sample Theory of Estimation[J]. Biometrika, 1973, 60(2): 457-465.
[3] BARTH D J. The Sampling Distribution of an Estimate Arising in Life Testing[J]. Technometrics, 1963, 3(5): 361-374.
[4] BARLOW E, PROSCHAN F. Mathematical Theory of Reliability[C]. New York Wiley, 1965,320-322.
[5] GUO S X. Efficient method for computing the fault probability of large-scale network system [J]. Systems Engineering and Electronics, 2005, 25(4): 744-747.
[6] LALIT G, Yan O. Radial Distribution System Reliability Worth Evaluation Utilizing the Monte Carlo Simulation Technique[J]. Computers Electrical Engineering, 2001, 27: 273-285.
[7]程述汉,董厚奎.利用次序统计量估计种群的平均寿命[J].生物数学学报, 2000, 15(3): 120-126.
[8] BRYANT R E. Graph-based Algorithm for Boolean Function Manipulation[J]. IEEE Transaction on Comput, 1986, C-35: 677-691.
[9]陈景鹏,金星.不同指数单元的串联系统的平均寿命评定[C].第六届全国计算机应用联合学术会议论文集, 2002: 40-42.
[10]于丹.复杂系统可靠性分析中的若干统计问题与进展[J].系统科学与数学, 2007, 27(1): 68-81.
[11]金星.大型复杂系统平均寿命评定的Monte Carlo方法[J].系统仿真学报, 2005, 17(1): 66-68.
[12]张艳等.基于置信分布的系统可靠度评估蒙特卡罗方法[J].北京航空航天大学学报, 2006, 32(10): 1023-1025.
[13]刘颖.模糊随机不可修复系统的可靠度和平均寿命分析[D].天津:天津大学, 2006: 7-14.
[14] REPICI M, SORNIOTTI A. A General Approach to Study the Reliability ofComplex Systems[C]. Acta Polytechica, 2003, 43(3): 34-40.
[15]李学京,韩筱爽.复杂系统平均寿命综合评估方法研究[J].数学的实践与认识, 2007, 37(2): 110-113.
[16]曹晋华,程侃.可靠性数学引论[M].修订版.北京:高等教育出版社, 2006: 1-12.
[17]陈常顺.平均剩余寿命置信下限的评估[J].中北大学学报, 2007, 28(3): 208-211.
[18]李学京.可靠性综合验证方法研究[J].中国科学院文献情报中心, 2007, 3: 42-47.
[19]陈家鼎.生存分析与可靠性[M].北京:北京大学出版社, 2005.
[20] JANE C, YUAN J. A Sum of Disjoint Products Algorithm Network-reliability Evaluation of Flow Networks[J]. Reliability Engineering and System Safety, 1993, 131: 664-675.
[21]杨军,赵宇.复杂系统平均剩余寿命综合评估方法[J].航空学报, 2007, 28(6): 1351-1354.
[22]具光花,周艳魏.完全样本下威布尔分布平均寿命的评估[J].延边大学学报, 2008, 34(4): 246-249.
[23]周源泉,翁朝曦.可靠性评定[M].北京:科学技术出版社, 1990: 72-79.
[24]张国志.复杂系统可靠性分析[D].北京:北京工业大学, 2009:22-24.
[25] ALBERT H, SNEAD R C. A Comparison of Monte Carlo Technique for Obtaining System Reliability Confidence Limits From Component Test Data[J]. IEEE Transaction on Reliability, 2007, 1(20): 320-332.
[26] LEV V, UTKIN. Interval Reliability of Typical Systems with Partially Known Probabilities[R]. European Journal of Operational Research, 2004, 153(3): 790-802.
[27] FOTUHI F M, SENIOR M A. Novel Approach to Determine Minimal Tie-sets of Complex Network[J]. IEEE Transaction on Reliability, 2004, 53(1): 61-70.
[28]翟伟丽,茆诗松.定时截尾场合下双参数指数分布的参数估计[J].应用概率统计, 2002, 18(2): 197-200.
[29]高峰,温书.定时截尾样本下指数分布平均寿命的假设检验[J].淮阴师范学院学报, 2004, 3(1): 14-15.
[30]陈家鼎,王宏.关于定时截尾寿命试验方案下指数分布情形平均寿命的置信限[R].北京:北京大学, 1995.
[31]王宏.混合截尾寿命试验情形下指数分布平均寿命的置信限[J].应用概率统计, 1996, 12(4): 20-22.
[32] YEY W. The Extension of Universal Generating Function Method to Search for All One-to-many D-minimal Paths of Acyclic Multi-state-arc for Conserv-ation Networks[J]. IEEE Transactions on Reliability, 2008, 57(1): 94-102.
[33] WEI C. A Simple Heuristic Algorithm for Generating All Minimal Paths[J]. IEEE Transactions on Reliability, 2007, 56(3): 488-494.
[34] TANG J. Mechanical System Reliability Analysis Using a Combination of Graph Theory and Boolean Function[J]. Reliability Engineering and System Safety, 2001, 72 (1): 21-30.
[35]范金成.统计推断号引[M].北京:科学出版社, 2001.
[36] SINGH B. Inferences Concerning Exponential Distributions in the Presence of Randomly Right Censored Data with Missing Censored Values, Lifetime Data Analysis[J]. IEEE Transaction on Reliability, 2002, 12(8): 69-88.
[37] CHANG Y. OBDD-Based Evaluation of Reliability and Importance Measures for Multisate Systems Subject to Imperfect Fault Conerage[J]. IEEE transactions on dependable and secure computing, 2005, 2(4): 336-347.
[38]田庭,刘次华.定时截尾缺失数据下指数分布的统计推断[J].华侨大学学报, 2006, 27(1): 20-22.
[39]陈家鼎,房祥忠.任意序贯样本下指数分布均值置信限[J].中国科学(A辑), 2005, 35(4): 1-8.
[40]李林.多次混合截尾寿命试验指数分布平均寿命的统计推断[D].北京:北京大学, 2005: 12-26.
[41]闫霞,于丹,李国英.一类可修威布尔型设各可用度Fiducial推断[J].系统利学与数学, 2004, 24(1): 17-27.
[42]严广松,侯紫燕.不相同定时截尾试验中平均寿命的置信限[J].郑州纺织工程院学报, 1998, 9(3): 19-21.
[43]尹逊汝.Ⅰ型截尾模型修正后平均寿命的非参数估计[J].泰山学院学报, 2007, 29(3): 20-23.
[44] TANG J. Mechanical System Reliability Analysis Using a Combination of Graph Theory and Boolean Function[J]. Reliability Engineering and System Safety, 2001, 72 (1): 21-30.
[45]廖林生,邓文丽.区间数据样本下指数分布平均寿命的矩估计[J].江西科学, 2007, 25(5): 540-545.
[46]张民悦.复杂可修系统的一类可靠性数学基础[J].兰州理工大学学报, 2002, 28(3): 118-120.
[47]荆广珠,房祥忠.非等定时截尾寿命试验方案指数分布情形平均寿命的置信限[J].数理统计与管理, 2000, 04: 46-49.
[48]张定海,张悦民. n元件复杂系统的可靠性分析[J].甘肃科学学报, 2007, 19(2), 131-132.