n中取k系统的冗余分配与变异性研究

In the reliability theory text, a k-out-of-n system is a very popular structure of redundancies in fault-tolerant systems, which has widely applications in eletronic engineering, aerospace, and water conservancy and hydropower and other related areas. The allocation of active redundancies is usually carried out so as to optimize the lifetime of the coherent system. This project is mainly concentrated on the problem of stochastic comparisons of lifetimes of k-out-of-n systems under the active spare allocation at the component level and the active spare allocation at the system level with respect to failure rate and reversed failure rate. On the other hand, the problem on the variability of lifetime of k-out-of-n system is also considered in this project. To be specific, we shall discuss the problem of stochastic ordering of k-out-of-n systems having independent and heterogeneous exponentially distributed components with respect to the Lorenz order and the star order, which compare random variables according to both their “magnitude” and their “spread”. The study in this project would be of important significance to guide system design and maintenance strategy formulation.

在可靠性理论中，n中取k系统是一类非常流行的纠错结构，被广泛应用于电子工程、航空工业以及水利水电等相关行业中。通常工程师会通过分配冗余元件来强化系统的可靠度，而采取何种分配策略使得系统寿命最大化则是其中的焦点。本项目基于具有独立不同分布元件的n中取k系统，致力于研究元件水平和系统水平两种热分配策略在失效率以及反失效率意义下的优劣性。本项目同时也关注n中取k系统的变异性。通过利用刻画变量“变异性”的Lorenz序和星序，探讨两个具有独立不同指数分布元件的n中取k系统之间的序性质。本项目对指导系统设计和维修策略的制定有重要意义。

本项目中，我们针对n中取k系统研究了元件水平和系统水平两种热分配策略的优劣性以及系统的变异性问题。首先，我们将传统的冗余分配方式由并联结构一般化为m中取l结构。我们发现，当备件完全匹配时，元件水平下的热分配在似然比序意义下优于系统水平下的热分配；当备件部分匹配时，在元件水平下的热分配以反失效率序在并联系统中优于系统水平下的热分配，而在串联系统中元件水平下的热分配则以失效率序保持对系统水平下的热分配的优性。另一方面，本项目基于scale模型在星序意义下研究了元件的非齐次性对于n中取k系统的变异性影响。我们发现当元件寿命服从指数、广义Weibull、广义Gamma，半正态等分布时，n中取n系统与n中取1系统的变异程度随着元件寿命的差异增大而增大。