1. Overwiew of Various System Reliability Analysis Methods 2010. 01. 03. Kim Hyoung Ju 1

2. Contents References Introduction System Reliability Analysis Methods ▪ Reliability block diagram ▪ Reliability graph ▪ Markov chain ▪ Monte Carlo simulation ▪ Fault tree analysis Reliability Graph with General Gates Modeling of RGGG Quatification of RGGG Summary Furthur work 2

3. References [1] M.C.Kim, " Development of a Quantitative Safety Assessment Method for nuclear I&C System including human operators ", 2004 [2] M. C. Kim, P. H. Seong, " Reliability graph with general gates : An intuitive and practical method for system reliability analysis", 2002 [3] http://www.math.rutgers.edu/courses/338/coursenotes/chapter5.pdf [4] http://www.dartmouth.edu/~chance/teaching_aids/books_articles/ probability_book/Chapter11.pdf [5] http://www.reliabilityeducation.com/rbd.pdf 3

4. Introduction Several methods can be applied to the system reliability analysis ; reliability block diagram, Markov chain, Monte Carlo simulation, fault tree analysis, and reliability graph. Each method has its own characteristics. 4

5. Reliability Block Diagram RBD performs the system reliability and availability analyses on large and complex systems using block diagrams to show network relationships. The structure of the reliability block diagram defines the logical interaction of failures within a system that is required to maintain system operation. RDB is diagrammatic method for showing how component reliability contribute to the success of failures of a complex system. It is drawn as a series of blocks connected in parallel or series configration. 5

6. Reliability Block Diagram A parallel connection is used to show redundancy and joined by multiple links or paths from the Start Node to the End Node. A series connection is joined by one continuous link from the Start Node to the End Node. 6 [AND][OR]

7. Reliability Graph Reliability graph is composed of nodes and arcs. It makes a one-to-one match between the actual structure of the system and the system model. It is the most intuitive method for analyzing system reliability. It cannot be widely used for system reliability analysis due to limited expression power. 7

8. [Data delivery system from node A to node D] The system is successful if there exist at least one path from node A to node D. Node D needs the outputs from node B and node D. Reliability Graph 8

9. Markov Chain A Markov chain is a random process with the Markov property. It is a sequence of random variables X 1, X 2, X 3,... with the Markov property. Probabilistic behavior of a Markov chain is determined by the dependencies between X 1 and X 2, between X 2 and X 3, etc. The next state only depends on the current state but not on the past state. 9

10. Numbers in each state indicate the success and failure of five transmission lines in order of a AB, a AC, a CB, a BC, and a CD. Markov Chain 10 [Reliability graph][Markov chain]

11. Monte Carlo Simulation Monte Carlo simulation method is useful for modeling phenomena with significant uncertainty in inputs. It uses repeated sampling to determine the properties of some phenomena. It considers random sampling of probability distribution functions as model inputs to produce many possible outcomes instead of a few discrete scenarios. It is based on the use of random numbers and probability statistics to investigate problems. 11

12. The numbers in each parenthesis are the failure probability of the transmission line. The numbers in each string are the generated random numbersfor one realization. Monte Carlo Simulation 12 [Reliability graph] [Monte Carlo Simulation]

13. Fault Tree Analysis FTA is a failure analysis in which an undesired state of a system is analyzed using boolean logic to combine a series of lower-level events. Fault tree is used to analyze a single fault event. Only one event can be analyzed during a single fault tree. Fault tree analysis has expression power. But it is not an intuitive method. 13 [Reliability graph] [Fault Tree Analysis]

14. Reliability Graph with General Gates Fault tree analysisReliability graph Expression powerLimited expression power Not intuitive methodIntuitive method RGGG method 14

15. D G001 nAnA Reliability Graph with General Gates -Modeling n1n1 n2n2 n yAyA yCyC 15 nBnB yByB n1n1 n2n2 n K n C n1n1 n2n2 n n1n1 n2n2 n [OR] yDyD [AND] [k-out-of-n] [general purpose gate]

16. 16 n0n0 ntnt njnj a ij G nini Reliability Graph with General Gates -Quantification njnj ajaj ajaj Pr {n j as successful} = Pn j Pr {a j as successful} = Pa j

17. 17 OR AND K-out-of-n y A =f A (x 1, x 2, …, x n ) = x 1 ∨ … ∨ x n =y 1 w 1 ∨ … ∨ y n w n y B =f B (x 1, x 2, …, x n ) = x 1 ∧ … ∧ x n = y 1 w 1 ∧ … ∧ y n w n (1≤l≤n) (l=0) (l=n) (otherwise) y C =f C (x 1, x 2, …, x n ) = (x 1 ∧ … ∧ x n ) ∨ … ∨ (x n-k+1 ∧ … ∧ x n ) (l≥k) (l<k) l= the number of successful parent nodes among n P l (S) : Pr{y=1 | there are l successful parent nodes} P l (F) : Pr{y=0 | there are l successful parent nodes} = 1-P l (S) r j (l) : the reliability of w j (l) (j=1, 2, …,l) Reliability Graph with General Gates -Quantification

18. 18 Reliability Graph with General Gates -Simple example

19. Summary There are numerous methods for reliability analysis. Fault tree has expression power, it is not intuitive method. Whereas, reliability graph has limited expression power, it is intuitive method. RGGG was developed to analyze general gates using reliability graph. 19

20. Further work Study the reliability analysis methods. Bayesian Network Petri-Nets Find better methods to analyze system reliability. Find the way to improve the methods. 20

21. Thank You 21