Failure Probability of 2-within-Consecutive-( 2 , 2 )-out-of-( n , m ) : F System for Special Values of m

Problem statements: In this research, the researcher aimed to discuses the ystem failure probability of the model 2-within-consecutive (2, 2 ) out of (n, m) system for special values of m. Approach: The basic idea for evaluating the failure probabil ity was the usage of the number of configuration of k (k = 2, 3, 4) parallel columns e ach contained n components in a 2×2-matrix. Results: The equation for the linear k-within (r, s) out of (n, m) system were reached. In this study the failure probability of 2-within-consecutive (2, 2) out of (n, m) system for m = 2, 3, 4. Conclusion/Recommendations: In general, it was difficult to evaluate the failur e probability in the two-dimensional reliability structures such as the linear k-within (r, s) out of (n, m) system. The researcher established the failure probability and then the reliability of three special cases. It was recommended to generalize the results for any value s of k, r, s and m.


INTRODUCTION
The consecutive k-out-of-n: F system has been extensively studied in recent years [1][2][3] . The system is specified by n, the number of components and k, the number of consecutive failed components that lead to system failure. Generalizations of the consecutive-kout-of-n: F system has been reported in a considerable number of papers [4] . One of the generalizations is the linear connected-(r, s)-out-of-(n, m): F lattice system. It consists of mn components arranged in m rows and n columns. The system fails whenever there is at least one rectangle of dimension r×s which contains all failed components. Bounds, reliability evaluation and invariant optimal design of the linear connected-(r, s)out-of-(n, m): F lattice system is studied in [4][5][6] . Zuo, Lin and Wu [7] propose combined k-out-of-n: F, consecutivek-out-of-n: F and linear connected-(r, s)-out-of-(n, m): F lattice system structures and provide recursive formulas for the reliability of the combined system structures. Koutras [8] uses the Markov chain approach for reliability evaluation of Markov chain embeddable systems. He shows that the tool is very useful to a great variety of well-known one-dimensional reliability structures. However, it is very difficult or impossible to apply the idea to two-dimensional reliability structures such as the linear connected-(r, s)-out-of-(n, m): F lattice system and the above-mentioned combined system structures.
A further generalization of the linear connected-(r, s)-out-of-(n, m): F lattice system is the linear k-within (r, s)-out-of-(n, m): F lattice system. It consists of mn components arranged in m rows and n columns. The system fails whenever there is at least one cluster of size r × s, which contains k or more failed components. It becomes a linear connected-(r, s)-out-of-(n, m): F lattice system when k = rs. The linear k-within (r, s)out-of-(n, m): F lattice system is applied to Thin Film Transistor Liquid Crystal Display failure model. If XGA (1024×768 = total 786432 dot) TFT display system fails if and only if more than or equal to 10 dot fail in 10×10 dot matrix, then the system become to be linear 10-within (10, 10)-out-of (1024, 768): F lattice system [9] . Akiba and Yammoto [10] proposed approximate values of reliability of this system. This study gives a simple, direct combinatorial method for determining the system failure probability of the following models: 2-within consecutive-(2, 2)-out-of-(n, 2): F system 2-within consecutive-(2, 2)-out-of-(n, 3): F system 2-within consecutive-(2, 2)-out-of-(n, 4): F system Notation: p(n, k) Probability of system failure, k = 2, 3, 4 T k Tables with rows indexed by r = 0, 1, 2… and columns indexed by c = 0,1,2,…, k = 2, 3, 4 p Probability that a component functions q 1-p α(n, k, j) Number of configuration of k(k = 2, 3,4) parallel columns each contains n components having j total failures with at least two failed components in a 2×2-matrix C k (n, j) Entry in T k k = 2, 3,4 . The typical states of the system for n = 2, k = 2, 3, 4 will be shown in Fig. 1.

States
The states of the system are denoted by ij, where the first digit denotes the number of failed components in the first row and the second digit denotes the number of failed components in the second row Now , we denote the states in Fig. 1a by 10,20,and 11 respectively and denote the states in Fig. 1b by 12, 21 and 11 respectively, for example the state 12 means, the first row contains one failed component and two failed components in the second row.

Assumptions:
• Each component and the system, is either working or failed , the required numbers which end in 0 are enumerated by L 2 (n-1, j). Those end in 01, are enumerated by 2L 2 (n-2, j-1). The conclusion can be proved by induction: the hypothesis asserts that: L 2 (n-1, j) = C 2 (n-j, j) and L 2 (n-2, j-1) = C 2 (n-j, j-1) However, this implies: and since this is exactly the relation satisfied by the elements of T 2 , this is equivalent to C 2 (n-j+1, j): , the required numbers which end in 0 are enumerated by L 3 (n-1, j). Those end in 02, are enumerated by L 3 (n-2, j-2). Those end in 01, are enumerated by 3L 3 (n-2, j-1). Those end in 011, 0111,…

RESULTS AND DISCUSION
With ( ) n,k, j α as defined in the notation, number of the states which cause system failure, the probability of system failure is: To enumerate the states which cause system failure, we can evaluate, at first, the states at which the system is working and then we subtract this from kn ,k 2,3, 4 j A direct method for obtaining ( ) n, k, j α is by constructing table T k k = 2,3,4 as follows: Case 1: When k = 2, we construct T 2 .
Given n, from row 0 through row (n-1) and column 0 through column (n-1) of Table 1 T 2 as follows: • Entries of row r = 0 are: A one followed by zeros • Any entry in row r > 0 is the sum of the entry just above it and twice the entry immediate left neighbors, i.e.: The coefficients in p(n ,2) are: ( ) ( ) 2 2n n 1 C n j 1, j , j 2,3, j 2 n, 2, j 2n n 1 , j j 2 Therefore, p(n, 2) is complete determined by (1-3).
• a i multiple of entry immediate left neighbors to the entry 2) where a i = a i-1 +a i-2 with a 1 = 4, a 2 = 6 i.e.: Example 3: Suppose that n = 4; to find P(4,4), we construct

CONCLUSION
In this study, we study the failure probability of the model 2-Within consecutive-(2, 2)-out-of-(n, m): F system, for m = 2, 3, 4. We make exact simple formula of the failure probability in these cases. We construct Table 1-3 from which we get α(n, j, k) when m = 2, 3, 4.