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

多维召唤-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，我们给出了（r，s）-out-of-（m，n）：F格系统和二维k-within-continuous-（r，s）-out-of-（m，n）：F系统。在这项研究中，我们提出了3维系统作为一类召唤-k-系统。这些系统可应用于二维或三维空间的监控系统等。在本研究中，首先，我们提出了一种基于遗传算法或分支定界算法的n中取k：F召唤系统中元件优化排列的有效算法。对于中等规模的多维系统，给出了某些召唤k-系统可靠度的递推公式，并通过算法的阶数和数值实验结果证明了算法的有效性.对于大型系统，我们基于极限定理提出了系统可靠性的上下限或近似值。并通过数值实验证实了这些界或近似值的良好拟合性。进一步，利用上述基本思想，我们提出了网络系统性能指标或可靠性的递推算法。作为未来的工作，我们需要研究更复杂的系统，例如多状态的召唤-k-系统，以便研究真实的世界中更多的系统。

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.