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。

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.