Reliability Algorithms for Consecutive-k Systems

• Main Authors: Jen-Chun Chang, 張仁俊
• Other Authors: Rong-Jaye Chen
• Format: Others
• Language: en_US
• Published: 2000
• Online Access: http://ndltd.ncl.edu.tw/handle/36061694916231560147

博士 === 國立交通大學 === 資訊工程系 === 89 === Reliability algorithms are useful tools in reliability analyses and reliability optimizations. In this dissertation, we study and design efficient reliability algorithms for consecutive-k-out-of-n:F systems, weighted-consecutive-k-out-of-n:F systems, f-or-consecutive-k-out-of-n:F systems, f-within-consecutive-k-out-of-n:F systems, consecutive-k-out-of-n:F networks, consecutive-k-out-of-n:F flow networks, consecutive-k-r-out-of-n:DFM systems, and other reliability systems which do not have the “consecutive-k” property. We use two general approaches to develop our reliability algorithms: the first is the recursive equation approach, and second is the Markov chain approach. By carefully designed recursive equations and heterogeneous Markov chains, and under the supports of computation theory, automata theory, and sparse matrix data structures, our reliability algorithms are simpler and (or) more efficient than other published corresponding ones. In addition, we think that designing reliability algorithms case by case is a messy work. Therefore, we propose a “regular reliability model”. It is not a system, but a tool to specify the structures of various systems which may not have the “consecutive-k” property. When analyzing the reliability of a system, we first specify the system structure with the regular reliability model, and apply the automata theory to derive a minimal state heterogeneous Markov chain, then an efficient reliability algorithm can be obtained by implementing the Markov chain approach with the sparse matrix data structure. English Abstract ii Acknowledgements iii Contents iv 1. Introduction 1 1.1 Historical background 1 1.2 The problems and the methodologies 3 1.3 Outline of this dissertation 4 2. The consecutive-k-out-of-n:F system 6 2.1 Assumptions and notation 8 2.2 The linear consecutive-k-out-of-n:F system 10 2.2.1 Shanthikumar’s O(nk) time algorithm 11 2.2.2 Hwang’s O(n) time algorithm 12 2.2.3 A linear component replacement algorithm 13 2.3 The circular consecutive-k-out-of-n:F system 16 2.3.1 Hwang’s O(nk2) time algorithm 17 2.3.2 Antonopoulou and Papastavridis’s O(n2k) time algorithm 18 2.3.3 Wu and Chen’s O(nk) time algorithm 19 2.3.4 Hwang’s O(nk) time algorithm 20 2.3.5 Wu and Chen’s second O(nk) time algorithm 21 2.3.6 A simpler O(nk) time algorithm 22 2.3.7 A circular component replacement algorithm 25 3. The weighted-consecutive-k-out-of-n:F system 30 3.1 Assumptions and notation 32 3.2 The linear weighted-consecutive-k-out-of-n:F system 34 3.2.1 Wu and Chen’s O(n) time algorithm 35 3.3 The circular weighted-consecutive-k-out-of-n:F system 37 3.3.1 Wu and Chen’s incomplete O(min{n, k}·n) time algorithm 38 3.3.2 An O(Tn) time algorithm 41 4. The f-or-consecutive-k-out-of-n:F system ……………………………...…… 46 4.1 Assumptions and notation 47 4.2 Chang, Cui and Hwang’s O(f2k2n) time algorithm 49 4.3 An O(fkn) time algorithm 52 4.4 Another O(fkn) time algorithm 53 4.5 An O((fk)kn) time algorithm 55 5. The f-within-consecutive-k-out-of-n:F system 59 5.1 Assumptions and notation 61 5.2 Hwang and Wright’s O(23kn) time algorithm 62 5.3 An O( n) time algorithm 64 6. The consecutive-k-out-of-n:F network 76 6.1 Assumptions and notation 78 6.2 Chen, Hwang and Li’s algorithm for k = 2 80 6.3 An O(2kn) time algorithm 83 7. The consecutive-k-out-of-n:F flow network 91 7.1 Assumptions and notation 92 7.2 An O( n) time f-flow-reliability algorithm 93 7.3 An O(k) time on-line routing algorithm 96 8. The consecutive-k-r-out-of-n:DFM system 99 8.1 Assumptions and notation 101 8.2 Koutras’s O((k+r)n) time algorithm 102 8.3 An O((k+r)n) time algorithm 103 8.4 An O(n) time algorithm 108 9. The regular reliability model 111 9.1 The regular reliability model 113 9.2 The F reliability model and the G reliability model 115 9.3 The relations among F models, G models, and regular models 118 9.4 An efficient reliability algorithm for the regular model 120 9.5 Applications 121 9.5.1 The f-or-consecutive-k:F model 121 9.5.2 The f-within-consecutive-k:F model 123 9.5.3 The k-mod-q model 125 9.5.4 Logic circuits 126 10. Conclusions 128 Bibliography 130 Vita 141 Publications 143