Evaluation and Design Method for Multi-Dimensional Consecutive-k-systems

多维连续k系统由排列在2维或3维空间中的组件组成，当狭窄空间中的许多组件失效时，该系统失效，但即使组件在宽空间中失效，该系统也不会失效。作为二维连续k-系统，我们有连通-(r, s)-out-of-(m, n): F晶格系统和二维k- in-连续-(r, s)-out-of-(m, n): F系统。在本研究中，我们提出了三维系统作为一种连续k系统。这些系统适用于二维或三维空间的监控系统。在本研究中，我们首先提出了基于遗传算法或分支定界算法的连续-k- of-n: F系统中组件最优排列的高效算法。对于中等规模的多维系统，提出了若干连续k-系统可靠度的递推公式，并通过算法顺序和数值实验结果证明了这些算法的有效性。对于大型系统，我们基于极限定理给出了系统可靠度的上界和下界或近似值。并通过数值实验验证了这些边界或近似值的良好拟合性。在此基础上，提出了网络系统性能指标或可靠性的递归算法。随着未来的发展，我们需要研究更复杂的系统，例如多状态连续k系统，以便在现实世界中研究更多的系统。

Multi-dimensional consecutive-k-systems consist of components arranged in a 2 or 3 dimensional space and fails when many components in a narrow space fail, but this system does not fail even if components fail in wide space. As the 2-dimensional oonsecutive-k-systems, we have connected-(r, s)-out-of-(m, n): F lattice system and 2-dimensional k-within-consecutive-(r, s)-out-of-(m, n) :F system. In this study, we proposed 3 dimensional systems as a type of consecutive-k-systems. These systems are applied to supervisory system for two or three dimensional space and so on.In this study, first, we proposed efficient algorithm for optimal arrangement of components in consecutive-k-out-of-n: F system, based on genetic algorithm or branch and bound algorithm. For multidimensional systems with medium size, recursive formulas for the reliabilities of some consecutive-k-systems were proposed and the efficiency of these algorithms were proven by the order of algorithm and the results of numerical experiments. For large systems, we proposed upper and lower bounds or approximate values of system reliabilities based on limit theorems. And we confirm the good fitness of these bounds or approximate values by numerical experiments. Furthermore, by using the above basic idea, we proposed recursive algorithm for performance index or reliability of network system.As the future works, we need to study more complex systems, for example, multi-states consecutive-k-systems, in order to research more systems in the real world.