元件寿命相依的多状态系统及其性质研究

This project is devoted to studying the stochastic properties of the multi-state systems with dependent components. Most of the studies in a multi-state setup are based on the assumption of s-independent components. However, in most of practical situations, the assumption is not reasonable. In this project, from the actual engineering background, the common environmental factors of components will be taken into account to model the dependence between components in a multi-state setup by proposing two important models, named the static environmental effect model and the dynamic environmental model. Moreover, we will investigate how the environment factors affect the performance of a multi-state system.. With the assumption of the dependence, we will further study the stochastic comparisons of the performances of systems and the evaluation of the stochastic behavior of survival over time. Stochastic comparisons of systems play a key role in building and optimizing systems. System signature is proved to be the most effective tool to compare two systems in binary context. For the purpose of studying the comparisons of two multi-state systems, we will propose the concept of signature in the context of multi-state systems with dependent components, and then establish some useful stochastic comparison results. As two of the most measures to evaluate the system ageing behavior, MRL and MIT have been widely investigated in binary context. However, in the multi-state setup, there has not been comprehensive and in-depth research on them. In view of the theoretical and practical points, in this project, we will propose the generalized MRL and MIT of multi-state systems, and investigate their properties such as monotonicity, asymptotic behavior etc for a more comprehensive evaluation of system ageing behaviors.

多状态系统是近年来系统可靠性研究的重心，但文献中绝大部分的研究都是在元件寿命独立的假定下进行的，这是不太合理的。本项目中我们从实际背景出发，考虑元件所处环境对其寿命的影响，建立静态和动态环境模型来描述元件的相依，并进一步研究环境对系统性能的影响机制。. 在元件相依的假定下，研究系统性能的随机比较和系统年龄行为的刻画。系统性能的比较对系统构建和优化都有着至关重要的作用。Signature是研究这类问题的有力工具。我们将扩展Signature概念至元件一般相依的多状态情形并借此研究多状态系统的性能比较。作为两个最重要的系统年龄行为的度量，平均剩余寿命(MRL)和休止时间(MIT)在两状态系统下被全面研究，但文献中关于多状态的MRL和MIT的研究却比较狭义和片面。本项目中我们考虑更广义且更符合实际背景的MRL和MIT，研究它们的性质如单调性和渐进行为对系统年龄行为的刻画。

本项目在元件寿命相依的多状态系统的随机性质方面展开深入的研究并取得了预期成果，基本完成了项目申请书中列出的目标。本项目三个方面研究目标均得到了较好的完成。得到以下重要结果：在两个模型多元Frailty模型和多元比例反失效率混合模型下，对常见系统的可靠性进行了研究；利用随机比较理论研究了环境的变化对系统性能指标的影响；动态模型下，在Marshall-Olkin 型冲击模型中，我们考虑了两种类型的冲击，得到了几个有关多元混合模型的相关结论。在多状态系统概率Signature与系统性能比较的研究中，提出了多状态系统联合signature的新的概念。利用这个新的概念，得到了系统的联合概率的表达，以及在实际计算系统signature的应用。对具有独立同分布以及独立不同分布元件的二状态n中取k系统进行了研究，并且推广到三状态单元件系统，进而推动了在多状态系统的广义MRL和MIT 的研究。