网络系统中可靠性问题的研究
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摘要
网络系统可靠性问题包括可靠性设计、可靠性分析、可靠性维护等一系列问题，其中网络可靠性分析是一个最基本的问题。网络可靠性分析一般是指给定网络部件可靠性参数的条件下，研究如何计算网络的可靠性。
由于信息网络、输电网络、集成电路网络、交通网络等网络广泛存在于现实世界，它们的正常运行与否不仅影响着普通大众的生活，也影响着一个国家的社会安全、经济发展等问题。因此网络可靠性问题不但是网络开发者和运营者关心的问题，更是学者们一直关注的课题。同时由于网络系统的复杂性，网络可靠性分析具有相当的难度，所以网络可靠性分析在方法上、理论上以及理论模型等许多方面还需要进行深入的研究。
本文从理论与方法两个方面对网络系统中有关可靠性问题进行了较为深入的研究。研究内容主要包括四个部分：网络可靠性界的计算、网络综合可靠性分析方法的研究、马尔可夫型可修网络系统中聚合问题的研究以及网络的模糊可靠性分析方法的研究。具体内容如下：
●简单介绍了网络可靠性问题的背景，研究现状、意义以及网络可靠性研究所涉及的数学方法。
●给出边数一定的网络断集数目的计算方法，数据表明其有效性。
●通过研究网络的连通子网络数与网络断集数目的关系，给出网络全端可靠性多项式系数界的计算公式，从而得到网络全端可靠性界的计算公式。实例表明所给的界对精确值具有较好的近似。
●利用网络最大概率状态的思想，在网络部件具有相同可靠度时，给出网络分析中所需状态包含的最大故障部件数目，获得全端可靠性界的又一种计算方法，数值结果比较表明所给出的界优于Jacobos，BBST，Kruskal-Katona,Ball-Provan．界。
●借助边变量将网络表示为一个代数系统，然后将k端可靠性问题进行转化后给出其上界的计算，并以Red Arpanet给出数值比较。
●提出用部件的稳态可用度生成网络的最大概率状态空间，借助融合顶点法判断生成状态的正常与否，由此可利用马尔可夫理论获得可修网络系统的一些重要可靠性指标，包括稳态可用性，首次故障前平均时间，故障频率等。
.以网络的业务性能作为网络状态正常与否的标准，建立了网络性能可靠性分
析的马尔可夫模型以及性能可靠性指标。并以公共信道信令网络的性能可靠性分
析为例，给出相关结果，数据显示了方法的合理性与正确性。
.给出容量相关的可靠性简约规则以及网络路径函数的矩阵生成方法，借助有
序二元决策图的性质解决了容量的计算问题，从而得到容量相关可靠性的一种计
算方法。
.针对网络可靠性问题中状态空间数目巨大的问题，研究了网络可靠性的连续
时间马尔可夫模型的聚合问题，通过研究聚合性与转移率矩阵之间的关系，给出
了用转移率矩阵判断可聚合性与几乎可聚合的条件，包括聚合的充要条件、必要
条件以及几乎可聚合的充分条件。
在Profust(基于概率与模糊态假设)领域内，在网络的最大概率状态空间内，
给出了可修网络系统的模糊可用性计算公式。给出了马尔可夫型可修网络系统中
模糊可靠性稳态指标。研究了文献中两种模糊可靠度之间的关系，指出其区别与
适用性，并以发射系统的模糊可靠性分析为例予以说明。给出了连续恤(F)系
统的模糊可靠性的分析与计算。
关键词:网络系统可靠性可用性公共信道信令网络马尔可夫模型
聚合profust可靠性连续协(F)系统
Network reliability encompasses a range of issues related to the design, analysis and calculate, maintenance of network, one of the most basic is network reliability analysis, which is devoted to research how to calculate the reliability of a network subject to their components reliability parameter.Because of widely presenting of Computer network, Communication network, transmission network and traffic network etc. in realistic world, whether they can operate stability or not influence not only the life of common population but also safety and stabilize of community, economic development etc of a country. So Network reliability problems much more than only concern of network developer and operator, especially is a hot topic paid close attention by researchers. At the same time network reliability problems have the definite trouble, therefore there are a lot of issues include the methods of analysis, theory models etc. need to be studied deeply.From analysis methods and theory models this paper studied some network reliability issues exist in network systems. The research is comprised mainly of four aspects: bounds of network reliability; performability of network reliability; lumpability problems of network reliability models; fuzzy analysis methods of network reliability. The main results are listed in the following:· The background, status, significance and mathematic methods of involving of network reliability are introduced briefly.· An algorithm of network cutest number with certain edges is given and numerical examples demonstrated the feasible and efficiency of the algorithm.· After studied the relation between connected subnet numbers and the cutest numbers of the network, a formula for computing the coefficient bounds of all-terminal reliability polynomial is obtained, then obtain the bounds of all-terminal reliability. The good performance of the bounds is illustrated by comparing the accurate value and bounds value of a SDH network.· Another bound of all-terminate reliability based on most probable strategy is given in the case of network components with same reliability. Numerical example
demonstrates the bound is superior to Jacobos, BBST, Kruskal-Katona, Ball-Provan.·Bound of k-terminate reliability is presented by using the edges variable to obtain the algebraic represent of network, then convert k-terminate reliability problem into another problem. An example of Red Arpanet is given to comparison the results.· Proposed a method by using components' steady availability to generate most probable state space of network and merging-vertex algorithm to judge whether a state connected or not. Then Markov theory is used successfully and some important reliability indexes of network system are obtained including steady availability, mean time to first failure, steady fault-frequency.· Established a performability model of network system under the performance criterion, and used Markov theory obtained some performance indexes. An example cf performability analysis of a common channel signaling network (CGSN) under delay constraint demonstrated the feasible and efficiency.· Presented a reduction axiom of capacity related network and matrix reduction method for obtaining path function of the network. A new method for calculating the capacity reliability is given by means of ordered binary decision diagram of Boolean function.· Investigated the lumpability problems of continuous time Markov model of network system, analysis the relationships between lumpable, nearly lumpable and generator matrix and the conduction satisfied by generator matrix under which Markovian process is lumpable, nearly lumpable is given.· In the field of profust theory, profust availability of repairable network system is present; profust reliability indexes of Markov model is given; The relationships between two kind of definition of fuzzy reliability existed in literatures is analysied. Profust reliability analysis of Consecutive-k-out of-n:F system and a emit system are given.
由于信息网络、输电网络、集成电路网络、交通网络等网络广泛存在于现实世界，它们的正常运行与否不仅影响着普通大众的生活，也影响着一个国家的社会安全、经济发展等问题。因此网络可靠性问题不但是网络开发者和运营者关心的问题，更是学者们一直关注的课题。同时由于网络系统的复杂性，网络可靠性分析具有相当的难度，所以网络可靠性分析在方法上、理论上以及理论模型等许多方面还需要进行深入的研究。
本文从理论与方法两个方面对网络系统中有关可靠性问题进行了较为深入的研究。研究内容主要包括四个部分：网络可靠性界的计算、网络综合可靠性分析方法的研究、马尔可夫型可修网络系统中聚合问题的研究以及网络的模糊可靠性分析方法的研究。具体内容如下：
●简单介绍了网络可靠性问题的背景，研究现状、意义以及网络可靠性研究所涉及的数学方法。
●给出边数一定的网络断集数目的计算方法，数据表明其有效性。
●通过研究网络的连通子网络数与网络断集数目的关系，给出网络全端可靠性多项式系数界的计算公式，从而得到网络全端可靠性界的计算公式。实例表明所给的界对精确值具有较好的近似。
●利用网络最大概率状态的思想，在网络部件具有相同可靠度时，给出网络分析中所需状态包含的最大故障部件数目，获得全端可靠性界的又一种计算方法，数值结果比较表明所给出的界优于Jacobos，BBST，Kruskal-Katona,Ball-Provan．界。
●借助边变量将网络表示为一个代数系统，然后将k端可靠性问题进行转化后给出其上界的计算，并以Red Arpanet给出数值比较。
●提出用部件的稳态可用度生成网络的最大概率状态空间，借助融合顶点法判断生成状态的正常与否，由此可利用马尔可夫理论获得可修网络系统的一些重要可靠性指标，包括稳态可用性，首次故障前平均时间，故障频率等。
.以网络的业务性能作为网络状态正常与否的标准，建立了网络性能可靠性分
析的马尔可夫模型以及性能可靠性指标。并以公共信道信令网络的性能可靠性分
析为例，给出相关结果，数据显示了方法的合理性与正确性。
.给出容量相关的可靠性简约规则以及网络路径函数的矩阵生成方法，借助有
序二元决策图的性质解决了容量的计算问题，从而得到容量相关可靠性的一种计
算方法。
.针对网络可靠性问题中状态空间数目巨大的问题，研究了网络可靠性的连续
时间马尔可夫模型的聚合问题，通过研究聚合性与转移率矩阵之间的关系，给出
了用转移率矩阵判断可聚合性与几乎可聚合的条件，包括聚合的充要条件、必要
条件以及几乎可聚合的充分条件。
在Profust(基于概率与模糊态假设)领域内，在网络的最大概率状态空间内，
给出了可修网络系统的模糊可用性计算公式。给出了马尔可夫型可修网络系统中
模糊可靠性稳态指标。研究了文献中两种模糊可靠度之间的关系，指出其区别与
适用性，并以发射系统的模糊可靠性分析为例予以说明。给出了连续恤(F)系
统的模糊可靠性的分析与计算。
关键词:网络系统可靠性可用性公共信道信令网络马尔可夫模型
聚合profust可靠性连续协(F)系统
Network reliability encompasses a range of issues related to the design, analysis and calculate, maintenance of network, one of the most basic is network reliability analysis, which is devoted to research how to calculate the reliability of a network subject to their components reliability parameter.Because of widely presenting of Computer network, Communication network, transmission network and traffic network etc. in realistic world, whether they can operate stability or not influence not only the life of common population but also safety and stabilize of community, economic development etc of a country. So Network reliability problems much more than only concern of network developer and operator, especially is a hot topic paid close attention by researchers. At the same time network reliability problems have the definite trouble, therefore there are a lot of issues include the methods of analysis, theory models etc. need to be studied deeply.From analysis methods and theory models this paper studied some network reliability issues exist in network systems. The research is comprised mainly of four aspects: bounds of network reliability; performability of network reliability; lumpability problems of network reliability models; fuzzy analysis methods of network reliability. The main results are listed in the following:· The background, status, significance and mathematic methods of involving of network reliability are introduced briefly.· An algorithm of network cutest number with certain edges is given and numerical examples demonstrated the feasible and efficiency of the algorithm.· After studied the relation between connected subnet numbers and the cutest numbers of the network, a formula for computing the coefficient bounds of all-terminal reliability polynomial is obtained, then obtain the bounds of all-terminal reliability. The good performance of the bounds is illustrated by comparing the accurate value and bounds value of a SDH network.· Another bound of all-terminate reliability based on most probable strategy is given in the case of network components with same reliability. Numerical example
demonstrates the bound is superior to Jacobos, BBST, Kruskal-Katona, Ball-Provan.·Bound of k-terminate reliability is presented by using the edges variable to obtain the algebraic represent of network, then convert k-terminate reliability problem into another problem. An example of Red Arpanet is given to comparison the results.· Proposed a method by using components' steady availability to generate most probable state space of network and merging-vertex algorithm to judge whether a state connected or not. Then Markov theory is used successfully and some important reliability indexes of network system are obtained including steady availability, mean time to first failure, steady fault-frequency.· Established a performability model of network system under the performance criterion, and used Markov theory obtained some performance indexes. An example cf performability analysis of a common channel signaling network (CGSN) under delay constraint demonstrated the feasible and efficiency.· Presented a reduction axiom of capacity related network and matrix reduction method for obtaining path function of the network. A new method for calculating the capacity reliability is given by means of ordered binary decision diagram of Boolean function.· Investigated the lumpability problems of continuous time Markov model of network system, analysis the relationships between lumpable, nearly lumpable and generator matrix and the conduction satisfied by generator matrix under which Markovian process is lumpable, nearly lumpable is given.· In the field of profust theory, profust availability of repairable network system is present; profust reliability indexes of Markov model is given; The relationships between two kind of definition of fuzzy reliability existed in literatures is analysied. Profust reliability analysis of Consecutive-k-out of-n:F system and a emit system are given.
引文
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