Mathematical Sciences: Stochastic Models for Reliability of Systems with Dependencies Among Components

建议：DMS 9503104 PI：W.J.Padgett，J.D.Lynch和S.D.Durham Institution：南卡罗来纳大学哥伦比亚分校标题：具有部件相关性的系统可靠性的随机模型摘要：这项研究涉及复杂系统的可靠性。主要有三个方面：(1)建立部件可靠性模型；(2)将部件可靠性和依赖关系纳入系统可靠性中，从而得到易于处理的数据分析模型；(3)为此类模型建立关键程度理论。针对(1)，研究了几种用于构件可靠性分析的模型，包括条件威布尔分布和逆高斯分布以及泊松-威布尔缺陷模型。研究了“条件威布尔”分布，因为它很好地符合某些纤维强度数据集，并且可能由于纤维制造和测试过程中的“审查”考虑而被证明是合理的。具有有限状态马尔可夫随机强度的泊松-威布尔缺陷模型以混合分布(零强度)和混合危险模型(无限强度)为极值。该项目这一部分的一个主要目标是为这一缺陷模型研究统一的混合理论，该模型包含了分别为混合分布和混合危险的极端情况开发的混合理论。为了解决(2)和(3)，我们研究了一个具有层次性的一般模型。该层次结构由(I)微观(或组件)级、(Ii)子系统(或“组件束”)级和(Iii)系统(或“束链”)级组成。人们早就知道，在许多系统中，一个组件的故障会改变施加在其余组件上的应力。因此，为了获得准确的结果，以一种现实的方式将组件依赖项合并到可靠性模型中是非常必要的。在这里，组件依赖关系/相互作用通过使用“负载分担规则”被合并到模型中，该规则最适用于“负载”是由于机械或物理考虑的情况。考虑了一般的单调负荷分担规则，给出了系统可靠性的计算方法。本研究包括一类特殊的这些规则，其中组件负载可以使用网络上随机游动的吸收概率来计算。特别是，一些流行的负荷分担规则可以归结为对电力网络的考虑。能量方面的考虑因素使我们能够深入了解部件失效时由这些规则引起的应力集中行为。主要目标是继续研究这些电气网络规则，以及其他适用于机械载荷传递情况的网络规则，特别是复合材料，特别是研究载荷可以通过基质材料传递的“有效距离”及其与通过基质传递载荷的纤维断裂周围的“无效长度”的关系。关键问题也被考虑在内。对于n中取k系统和具有少量部件的系统的初步工作表明，当每个部件具有威布尔失效分布时，使用混合变换伽马模型可以得到精确的理论，其中变换依赖于威布尔分布。这种模型也适用于极值逼近误差的计算和使用混合分布的系统辨识。这项研究涉及开发描述具有相关部件的复杂系统的故障的模型。研究人员正在研究复杂部件系统的可靠性模型，其中包括允许部件依赖和相互作用的复杂材料。这些模型直接应用于许多当前重要的问题，包括纤维复合材料和电力网络的失效。这对高可靠性的复杂材料的设计和大规模制造具有重要意义。

Proposal: DMS 9503104 PIs: W. J. Padgett, J. D. Lynch and S. D. Durham Institution: University of South Carolina - Columbia Title: STOCHASTIC MODELS FOR RELIABILITY OF SYSTEMS WITH DEPENDENCIES AMONG COMPONENTS Abstract: The research involves the reliability of complex systems. There are three main thrusts: (1) modeling component reliability, (2) incorporating component reliability and dependencies into the system reliability which result in tractable data analytic models, and (3) developing a criticality theory for such models. Regarding (1), some models are studied for the analysis of component reliability, including the conditional Weibull and inverse Gaussian distributions and a Poisson-Weibull flaw model. The "conditional Weibull" distribution is studied since it fits certain fiber strength data sets well and may be justified due to "censoring" considerations in the fiber manufacturing and testing processes. The Poisson-Weibull flaw model with finite-state Markov random intensity has the mixed distribution (zero intensity) and the mixed hazard model (infinite intensity) as extremes. A major objective of this part of the project is the investigation of a unified mixture theory for this flaw model which subsumes the theory of mixtures developed for the extremes of mixed distributions and mixed hazards, respectively. To address (2) and (3), a general model is investigated which is hierarchical in nature. The hierarchy consists of (i) the micro (or component) level, (ii) the subsystem (or "bundle of components") level, and (iii) the system (or "chain of bundles") level. It has long been known that in many systems, the failure of a component changes the stress applied to the remaining components. Thus, incorporating component dependencies into a reliability model in a realistic manner is highly desirable for accurate results. Here, component dependencies/interactions are incorporated into the model by using "load-sharing rules," m ost applicable to situations where "loadings" are due to mechanical or physical considerations. General monotone load-sharing rules are considered for which a method of calculating the system reliability has been developed. The present research includes a special class of these rules where the component load can be calculated using absorption probabilities for random walks on a network. In particular, a number of the popular load-sharing rules can be reduced to the consideration of electrical networks. Energy considerations give insight into the behavior of stress concentrations induced by these rules as components fail. Major objectives are to continue the investigation of these electrical network rules, and other network rules which are appropriate for mechanical load transfer situations, especially for composite materials, and, specifically to study the "effective distance" that a load can be transferred via matrix material and its relationship to the "ineffective length" around fiber breaks for load transfer through the matrix. Criticality issues are also considered. Preliminary work for k-out-of-n systems and systems with a small number of components suggest that an exact theory may be obtainable when each component has a Weibull failure distribution using a mixed transformed gamma model, where the transformation depends on the Weibull distribution. Such a model also lends itself to calculation of extreme value approximation errors and to system identification using the mixing distribution. The research involves the development of models for describing the failure of complex systems with dependent components. The researchers are investigating reliability models for complex systems of components including complex materials which allow for component dependencies and interactions. Such models have direct application to many problems of current importance, including failure of fibrous composite materials and electrical networks. This has important imp lications for the design and large scale manufacture of complex materials where high reliability.