A 2-dimensional rectangular k-within-consecutive-(r,s)-out-of-(m,n):F system is a 2-dimensional version of a consecutive-k-out-of-n:F system. This system consists of m×n components, and fails if and only if a number of components more than or equal to k in any r×s submatrix fail. A 2-dimensional rectangular k-within-consecutive-(r,s)-out-of-(m,n):F system can be treated as a reliability model for TFT Liquid Crystal Display, wireless communication network and so on. An efficient method has been sought for evaluating the system reliability. Though the method is known to be effective for small or medium size systems, the method needs more computation time and more memory capacity, as the system size becomes larger. Therefore, we propose new formulas for upper and lower bounds and approximate value for system reliability, when the system size is large. In this paper, first, we propose new formulas for upper and lower bounds for reliability of the 2-dimensional rectangular k-within-consecutive-(r,s)-out-of-(m,n):F system. We executed some numerical experiments in order to compare our new proposed upper and lower bounds with ones proposed in the preceding studies. According to the results, when the system size is large, our proposed new lower bound is the best lower bound for all the systems. For upper bounds, it was shown that best upper bound depended on the size of matrix r×s, the number of failure components k, and component reliability. When system size is larger than m=50 and n=50, k is much smaller than r×s and component reliability is nearly equal to one, our proposed new upper bound is the best upper bound. Secondly, we propose a limit theorem for the 2-dimensional rectangular k-within-consecutive-(r,s)-out-of-(m,n):F system. This theorem gives us approximate reliabilities of the 2-dimensional rectangular k-within-consecutive-(r,s)-out-of-(m,n):F system, when the system size is large and component reliabilities are nearer to one.
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