本研究以線性連續的k-within-r-out-of-n:F系統可靠度為題材，此系統至今已廣泛的被許多學者研究討論及運用，且近年來相關研究也逐漸由基本的一維度延伸至高維度的可靠度系統，而本文主要是研究一維系統可靠度。首先，將各個元件以虛擬變數表示(成功為0、失敗為1)，接著針對相同元件系統之遞迴法與矩陣法做比較，研究何種方法在可靠度的計算上更為精簡、更有效率。

Computational method for the reliability of a linearly consecutive k-within-r-out-of-n:F system is studied in this thesis. We state first the assumptions about the manufacturing process of a production line system. We provide a simple and intuitive way to obtain the transition probability matrix of a finite Markov chain describing the system status over time. System reliability can then be obtained via the Markov chain. Numerical results and illustrations are given to show some characteristics of the system.

[1] Antonopoulou, I. and Papastavridis, S. (1987). Fast recursive algorithm to evaluate the reliability of a circular consecutive-k-out-of-n:F system, IEEE Transactions on Reliability, R-36, pp. 83-84.
[2] Boehme, T. K., Kossow, A. and Preuss, W. (1992). A generalization of consecutive-k-out-of-n:F systems, IEEE Transactions on Reliability, 41, pp. 451-457.
[3] Chang, Y. M. and Huang, T. H. (2010). Reliability of a 2-dimensional k-within-consecutive-r×s-out-of-m×n:F system using finite Markov chains, IEEE Transactions on Reliability, 59, pp. 725-733.
[4] Chang, Y. M. and Wu, T. L. (2011). On average run lengths of control charts for autocorrelated processes, Methodology and Computing in Applied Probability, 13, pp. 419-431.
[5] Chao, M.T. and Fu, J.C. (1989). A limit theorem of certain repairable systems. Annals of the Institute of Statistical Mathematics, 41, pp. 809–818.
[6] Chao, M.T. and Fu, J.C. (1991). The reliability of large series system under Markov structure. Advances in Applied Probability, 23, pp. 894–908.
[7] Chiang, D. T. and Niu, S-C (1981). Reliability of consecutive-k-out-of-n:F system, IEEE Transactions on Reliability, R-30, pp. 87-89.
[8] Salvia, A. A. and Lasher, W. C. (1990). 2-dimensional consecutive-k-out-of-n:F models, IEEE Transactions on Reliability, 39, pp. 382-385.
[9] Fu, J. C. and Chang, Y.M. (2002). On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials, Journal of Applied Probability, 39, pp. 70-80.
[10] Fu, J. C. (1986). Bounds for reliability of large consecutive-k-out-of-n:F systems with unequal component reliability, IEEE Transactions on Reliability, 35, pp. 316-319.
[11] Fu, J. C. (1986). Reliability of consecutive-k-out-of-n:F systems with (k-1)-step Markov dependence, IEEE Transactions on Reliability, R-35, pp. 602-606.
[12] Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach, Journal of the American Statistical Association, 89, pp. 1050-1058.
[13] Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials, Statistica Sinica, 6, 957-974.
[14] Kontoleon, J. M. (1980). Reliability determination of a r-successive-out-of-n:F system, IEEE Transactions on Reliability, R-29, p. 437.
[15] Lou, W.Y. W. (1996). On runs and longest run tests: method of finite Markov chain imbedding, Journal of the American Statistical Association, 91, 1595–1601.
[16] Lou, W.Y. W. and Fu, J. C. (2007). On exact Type I and Type II errors of Cochran’s test, Statistics and Probability Letters, 77, pp. 1282-1287.
[17] Markov, A. A. (1907). Issledovanie zamechatel’nogo sluchaya zavisimyh ispytanij [Investigation of a remarkable example of dependent trials], Izvestiya Petersburgskoi akademii nauk (6), 1, pp. 61-80.
[18] Yamamoto, H. and Miyakawa, M. (1995). Reliability of a linear connected-(r,s)-out-of-(m,n):F lattice system, IEEE Transactions on Reliability, 44, pp. 333–336.
[19] Zuo, M. (1993). Reliability and design of 2-dimensional consecutive-k-out-of-n systems, IEEE Transactions on Reliability, 42, pp. 488-490.