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1 ECE454/544: Fault-Tolerant Computing & Reliability Engineering Administrative Issues ECE544 project meeting by Oct. 28, Friday My availability: Thu: 12:30-3pm Fri: before 10am, 12-1:30pm Lecture #13 Binary Decision Diagrams (BDD) Instructor: Dr. Liudong Xing Fall 2016 Homework #5 assigned Due by Nov. 3, Thursday This class Finish L#12 (RBD Cont d) Then L#13 (BDD) Dr. Xing 2 Lecture #13 1

2 Review of Lecture #12 Topics A RBD is a success-oriented network describing the function of the system In terms of modeling capability, RBD is equivalent to the static/traditional fault trees, and they can be converted into each other easily Path-sets and cut-sets can be generated from both RBD and fault trees I/E and SDP can be applied to the quantitative analysis based on both path sets and cut sets System analysis can be performed based on structure functions Binary decision diagrams (BDD) Basic concepts BDD manipulation & construction Variable ordering in BDD Calculating (un)-reliability from the BDD Dr. Xing 3 Dr. Xing 4 Lecture #13 2

3 Shannon Decomposition & ite Format Let f be a Boolean expression and x be a Boolean variable: Binary Decision Diagrams (BDD) A BDD is a directed acyclic graph (DAG) based on Shannon decomposition f x=1 represents the function f evaluated at x=1 f x=0 represents the function f evaluated at x=0 The if-then-else (ite) format of the Shannon decomposition else edge /0-edge x F then edge /1-edge A BDD is a binary tree two sink nodes labeled with constants 1 and 0 each non-sink node is labeled with a Boolean variable x and has two outgoing edges called 1-edge (then-edge) and 0- edge (else-edge), respectively F x=0 F x=1 else edge /0-edge x F then edge /1-edge F x=0 F x=1 Dr. Xing 5 Dr. Xing Lecture #13 6 Lecture #13 3

4 Other Forms An ordered BDD (OBDD) is a BDD with the constraint that the variables are ordered and every source-to-sink path in the OBDD visits the variables in ascending order An reduced OBDD (ROBDD) is an OBDD where each node represents a distinct Boolean expression ROBDD ROBDD are widely used in practice! In reliability engineering: Sink node 1: the system fails Sink node 0: the system is functioning x=1: the component x is failed x=0: the component x is functioning BDD of computer system Dr. Xing 7 Dr. Xing Lecture #13 8 Lecture #13 4

5 Agenda BDD Construction (1) Binary decision diagrams (BDD) Basic concepts BDD manipulation & construction Variable ordering in BDD Calculating (un)-reliability from the BDD To generate ROBDD, the ordering of the variables (basic events) must be selected An index is assigned to each variable to indicate its position in the ordering The order is not changed during the generation Index(x) < index (y) implies that y is behind x in the order of variables Dr. Xing Lecture #13 9 Dr. Xing Lecture #13 10 Lecture #13 5

6 BDD Construction (2) BDD is constructed from the bottomup The BDD of a basic event / component: BDD Construction (3) AND ing two BDD: index (x) < index (y) X Y AND X does not occur The "system" represented by X does not fail X occurs Dr. Xing Lecture #13 11 X ite(x, 1, 0) The "system" represented by X fails First choose the "lesser" variable to be the root X Y AND AND Y Then, AND Y with the child nodes of X (recursively) Dr. Xing Lecture #13 12 X 0 Y Assume: g = ite(x,1,0), h = ite(y,1,0) if index(x) < index(y), then g AND h = ite(x,1,0) AND ite(y,1,0) = ite(x, 1 AND h, 0 AND h) = ite(x, h, 0) Lecture #13 6

7 BDD Construction (4) OR ing two BDD: index (x) < index (y) Combination Operation on BDD Let Boolean expression g and h be: X OR Y First choose the "lesser" variable to be the root X X A logic operation (AND/OR, ) between g and h can be represented by the following BDD manipulation: Y OR OR Y Then, OR Y with the child nodes of X (recursively) Y 1 Assume: g = ite(x,1,0), h = ite(y,1,0) if index(x) < index(y), then g AND h = ite(x,1,0) OR ite(y,1,0) = ite(x, 1 OR h, 0 OR h) = ite(x, 1, h) Dr. Xing Lecture #13 13 The same rule can be used for logic operation between sub-expressions until one of them becomes a constant 0 or 1 Dr. Xing Lecture #13 14 Lecture #13 7

8 Reduction Rules To achieve ROBDD, two reductions are performed as the BDD is built. These ensure that the BDD that results is minimal for the chosen ordering Merge isomorphic subtrees (if two identical sub-bdd result, at least one is superfluous) Hands-On Problem Generate the BDD for the following fault tree using variable ordering of A<B<C<D FT-BDD1 Delete useless nodes (A node with two equal children is useless and should be replaced with one of its children.) Dr. Xing Lecture #13 15 Dr. Xing Lecture #13 16 Lecture #13 8

9 Agenda Variable Ordering in BDD Binary decision diagrams (BDD) Basic concepts BDD manipulation & construction Variable ordering in BDD Calculating (un)-reliability from the BDD The size of BDD depends heavily on the input variable ordering used to build the BDD Dr. Xing Lecture #13 17 Dr. Xing Lecture #13 18 Lecture #13 9

10 Example Generate BDD using two different ordering for the following fault tree (FT-BDD2): Example (Cont d) BDD for the example fault tree: f ( a ( b c)) ( d ( b c) e) Dr. Xing Lecture #13 19 Dr. Xing Lecture #13 20 Lecture #13 10

11 Note! The poor ordering can significantly affect the size of the BDD, thus the reliability analysis solution time for large systems! Currently there is no rule-based means of determining the best way of ordering basic events for a given fault tree structure Fortunately heuristics can usually be used to find a reasonable variable ordering Example Heuristics H1: Top-down, left-right approach The indexing results results from a depthfirst left-most traversal of the fault tree Example: the order in which variables are visited for FT-BDD2 induces the ordering Dr. Xing Lecture #13 21 Dr. Xing 22 Lecture #13 11

12 Example Heuristics (Cont d) H2: works in 3 steps Associate each terminal variable with weight 1 and propagate these weights bottom-up through fault tree by associating to each intermediate variable the sum of weights of variables occurring in its definition Sort arguments of connectives in increasing order of their weights Apply H1 on the resulting fault tree H2: Example Example: consider fault tree FT-BDD2 Associate weight 1 to a,b,c,d,e Compute the weights of g0, g1, g2, g3 Rewrite g2 as g0, g1, g3 keep unchanged! Perform a depth-first, left-most traversal of the resorted fault tree Dr. Xing 23 Dr. Xing 24 Lecture #13 12

13 Example Heuristics (Cont d) H3: works in two steps Sort the arguments of each connective according to their number of references or fanouts Apply H1 on the resulting fault tree H3: Example Example: consider FT-BDD2 g1, g2 have only 1 fanout definition of g0 remain unchanged g3 has 2 fanouts (g1, g2), a,d,and e have 1 fanout g1, g2 are rewritten as b, c have 1 fanout definition of g3 remains unchanged! Visiting order resulted from a depth-first, left-most traversal of rewritten fault tree Dr. Xing 25 Dr. Xing 26 Lecture #13 13

14 Agenda Binary decision diagrams (BDD) Basic concepts BDD manipulation & construction Variable ordering in BDD Calculating (un)-reliability from the BDD Calculating Unreliability using BDD Each path through the BDD from the root to a leaf node represents a disjoint combination of component failures and non-failures A path with a leaf node labeled with a 1 leads to system failure Probabilities associated with arcs on each path are either q (component failure probability) for the right branch or (1-q) for the left branch System unreliability is given by the sum of the probabilities for all paths from the root to a leaf node labeled 1 Dr. Xing 27 Dr. Xing 28 Lecture #13 14

15 Consider FT-BDD1: Example Recursive BDD Evaluation Algorithm 0 A B C D 1 Consider a BDD branch G = ite(x, G 1, G 2 ) x G 2 G 1 Three paths from root (A) to leaf node labeled 1. Each path is disjoint by construction, so no subtraction is needed. Pr{system failure} = Pr{A and B}+ Pr{A and (not B) and C and D} + Pr{ (not A) and C and D} =q A q B +q A (1-q B ) q C q D +(1-q A ) q C q D If G=0 then P{G}=0 If G=1 then P{G}=1 Otherwise, P{G}=q x P{G 1 }+(1-q x )P{G 2 } When x is the root node of the BDD, P{G} gives the system unreliability. Dr. Xing 29 Dr. Xing 30 Lecture #13 15

16 Hands-On Problem (1) Find the BDD model for the following fault tree and evaluate the system reliability given that the failure probabilities of components are: Pr{A} = 0.01, Pr{B} = 0.05, Pr{C} = and that all failures are s-independent Review (Lecture#10) I-E Method Pr{F} = Pr{A + BC} = Pr{A} + Pr{BC} - Pr{ABC} = = A Washing Machine Overflows Fill mode too long SDP Method Pr{F} = Pr{A + BC} = Pr{A} + Pr{ABC} = *0.05*0.075 = B C Dr. Xing 31 Dr. Xing 32 Lecture #13 16

17 Hands-On Problem (2) For the fault tree called example FT2 in L#10, generate the BDD and calculate the probability of occurrence for the top event in the fault tree. Assume that the probability of occurrence for each of the basic events is: Pr{A} = 0.05, Pr{B} = 0.10, Pr{C} = 0.15, Pr{D} = 0.20, Pr{E} = 0.25 G1 BDD for Example FT2 A B D C E D G2 G3 G4 G5 E A B C D Dr. Xing 33 Dr. Xing 34 Lecture #13 17

18 References J. B. Dugan and S. A. Doyle. New Results in Fault-Tree Analysis Tutorial Notes of the Annual Reliability and Maintainability Symposium, January 1997 R. E. Bryant, Graph-based algorithms for boolean function manipulation, IEEE Trans. on Computers, vol. C-35, no. 8, pp , August A. Rauzy, New algorithms for fault tree analysis, Reliability Engineering and System Safety, vol. 40, pp , M. Bouissou, F. Bruyere, and A. Rauzy. BDD Based Fault-Tree Processing: A Comparison of Variable Ordering Heuristics, Proceedings of ESREL 97 conference, June Summary of Lecture #13 BDD can be used to efficiently and accurately solve the combinatorial reliability models without the use of cutsets The size of the BDD depends heavily on the input variable ordering Static fault trees, cutsets, pathsets, RBD, and BDD are combinatorial models! Next topic: Dynamic Fault Trees Dr. Xing 35 Dr. Xing 36 Lecture #13 18

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