1. Adding Dynamic Nodes to Reliability Graph with General Gates using Discrete-Time Method Lab Seminar Mar. 12th, 2007 Seung Ki, Shin

2. Korea Advanced Institute of Science and Technology Contents  Introduction  Dynamic Nodes  Probability tables for dynamic nodes  Discrete-time method  Conclusion  Further Study  References 1/15

3. Korea Advanced Institute of Science and Technology Introduction  Among the methods to analyze system reliability, fault tree analysis is the most widely used method, but it’s not an intuitive method.  Reliability graph with general gates (RGGG) method was proposed in order to overcome this shortcoming of fault tree.  Conventional fault tree and RGGG method cannot capture the dynamic behavior of system associated with time dependent events.  Dynamic fault tree method was proposed, but it’s also not intuitive. And when solving the reliability from dynamic fault tree, usually Markov chain method is used, but it has the state space explosion problem.  If it is possible to add a sequential concept to RGGG, it will become a intuitive dynamic method. 2/15

4. Korea Advanced Institute of Science and Technology Dynamic Nodes  Static RGGG nodes  (a) OR (b) AND (c) k-out-of-n(d) general purpose gate 3/15

5. Korea Advanced Institute of Science and Technology Dynamic Nodes  Additional dynamic nodes  Additional dynamic nodes for dynamic RGGG are proposed according to dynamic gates of dynamic fault tree. ① Priority-AND (PAND) gate Output occurs if and only if all input events occur in a particular order. The order of occurrence is the order in which the input events are connected to the PAND gate from left to right. AB 4/15

6. Korea Advanced Institute of Science and Technology Dynamic Nodes ② Spare (WSP, CSP, HSP) gate Spare gate is used to indicate that the output occurs if and only if all spare events (input) occur. ③ Functional Dependency (FDEP) gate All dependent events are forced to occur when the trigger event occurs. Primary active unit WSP 1st alternate unit 2nd alternate unit nth alternate unit … FDEP 1n … Trigger event Dependent Events 5/15

7. Korea Advanced Institute of Science and Technology Dynamic Nodes ④ Sequence Enforcing (SEQ) gate This gate forces events to occur in a particular order. The input events are constrained to occur in the left-to-right order in which they appear under the gate. SEQ 1 n … 2 6/15

8. Korea Advanced Institute of Science and Technology Probability Tables for Dynamic Nodes  Bayesian network  A probability-based method for representing and reasoning with uncertain knowledge. Each node in a BN is associated with a random variable. Links define probabilistic dependence relations between variables. The joint probability over the random variables in the network can be expressed as  Pa(x i ) is a configuration of the set of parents of variable X i  The first step in designing a BN is the definition of its graph. Then, the conditional probability of each node, given the values of its parents, must be assessed.  In order to transform a dynamic reliability graph to an equivalent Bayesian network, the probability table corresponding to each dynamic node should be determined. 7/15

9. Korea Advanced Institute of Science and Technology Probability Tables for Dynamic Nodes  Discrete-time method  Divide the process time line into n same intervals.  Output of each node is one of {I 1, I 2, …, I n, I ∞ }.  I k means that the node is failed in kth interval.  I ∞ means that the node is never failed. 8/15

10. Korea Advanced Institute of Science and Technology Probability Tables for Dynamic Nodes  Discrete-time method  P ij k : The probability that an arc a ij from node n i to n j is failed in kth time interval  When the cumulative failure distribution function of a ij is F ij (t),   i.e.) Exponential distribution 9/15

11. Korea Advanced Institute of Science and Technology Probability Tables for Dynamic Nodes  P-table for PAND node When n = 2 (Process time is divided into 2 intervals) a 1A a 2A  n A fails only when a signal from n 2 fails after a signal from n 1 fails. 10/15

12. Korea Advanced Institute of Science and Technology Probability Tables for Dynamic Nodes  P-table for WSP node (n = 2) a 1A a 2A  n A fails when both signals from main node(n 1 ) and spare node(n 2 ) fail.  P ij kα : the probability that an arc a ij is failed in kth time interval and at that time the arc a ij is in spare state.  α : dormancy factor which is the ratio of spare state failure rate to working state failure rate (CSP : α = 0, HSP : α = 1, WSP : 0< α<1) i.e.) failure rate of working state : λ failure rate of spare state : αλ 11/15

13. Korea Advanced Institute of Science and Technology Conclusion  If we use this method, the shortcoming of original RGGG that it can model only static system can be overcome.  As n increases, the obtained reliability becomes more exact.  If n is set as infinity, we can derive the ideally accurate reliability.  As n increases, the execution time for calculating system reliability increases seriously.  Discretization number, n should be determined by trade-off between the excution time and the solution accuracy. 12/15

14. Korea Advanced Institute of Science and Technology Conclusion  The system failure rate is usually very small, therefor n doesn’t have to be large in order to derive nearly exact reliability.  The probability that 2 components fails in same interval is very low.  A result of another paper about discrete-time method  It was shown that small values of n (from 1 to 5) are sufficient to get accurate system’s reliability. 13/15

15. Korea Advanced Institute of Science and Technology Further Study  Practical test on a certain dynamic system using this method.  Development of any other dynamic node.  Continuous-time method for more accurate result. 14/15

16. Korea Advanced Institute of Science and Technology References  M. C. Kim and P. H. Seong, “Reliability graph with general gates: an intuitive and practical method for system reliability analysis”, Reliability Engineering and System Safety, 2002.  J. B Dugan, S. J. Bavuso and M. A. Boyd, “Dynamic Fault-Tree Models for Fault- Tolerant Computer Systems”, IEEE Transactions on Reliability, 1992.  S. F. Galán and F. J. Diez, “Networks of probabilistic events in discrete time”, International Journal of Approximate Reasoning, 2002.  H. Boudali, J. B. Dugan, “A discrete-time Bayesian network reliability modeling and analysis framework”, Reliability Engineering and System Safety, 2005. 15/15