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1 Fault-Tolerant Computing Systems #6 Network Reliability Pattara Leelaprute Computer Engineering Department Kasetsart University pattara.l@ku.ac.th

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2 Network Network is made up of network component Network component Nodes Links (arcs, edges) connecting by HW or software component States of Network component Operational Failed

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3 Network Reliability Problems Input ： Probability that each component can operates normally Output ： Network Reliability Network Model Undirected graph G = (V, E) (V=vertices, E=edges) Edge ： operational or failed P e = Pr [edge e is operational] = reliability of e Unnecessary to think about time (=availability)

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4 Fault Model Situation of Network a c e b d v1v1 v2v2 v3v3 v4v4 p a =0.9 p b =0.8 p c =0.9 p d =0.9 p e =0.95 … P e = Pr [edge e is operational] = reliability of e

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5 Network Reliability k-terminal reliability Probability that there exist operating paths between every pair of nodes in K Two terminal reliability Probability that there exist operating path between 2 nodes (|K| = 2) All terminal reliability Probability that there exist operating paths between all nodes (K=V) K = set of nodes V = all nodes

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6 Minpaths Pathset A set of components (edges) whose operation implies (guarantees) system operation Minpath A minimal Pathset Ex ． K={v 1,v 4 } v1v1 v4v4

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7 Mincuts Cutset A set of components (edges) whose failure implies (guarantees) system failure Mincut A minimal Cutset Ex ． K={v 1,v 4 } v1v1 v4v4

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8 Computation of Reliability Complexity for two-terminal reliability and all terminal reliability NP-hard (#P-complete) Algorithms Efficient Algorithms for Restricted Classes Exponential time algorithm for general networks

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9 Transformations and Reductions R(G) = (multiplicative factor) * R(G’) G’ = contraction of G R(G) = reliability of G R(G’) = reliability of G’ Contraction G, G’ = (contraction of G, Ge) Multiplicative factor = p e When e is mandatory (mandatory = an edge that appears in every minpath) e uv u (= v) G:G:G’:

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10 Transformations and Reductions Parallel Reduction G, G’ Multiplicative factor = 1 Series Reduction G, G’ Multiplicative factor = 1 G:G:G’: p1p1 p2p2 p1 p2 p1p1 p2p2 1- (1- p1) (1- p2) G:G:G’:

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11 Series-Parallel Graphs A graph that can be contracted to one edge by using Series and Parallel Replacement Series Replacement Parallel Replacement There exists that algorithm to calculate K- terminal reliability in polynomial time.

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12 An Example Parallel Replacement Series Replacement

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13 An Example p1p1 p2p2 p1p1 p2p2 1- (1- p1) (1- p2) p1 p2 papa pbpb pcpc pdpd pepe papa pcpc pepe pbpdpbpd papa pcpc papa 1-(1-p e )(1- p b p d ) p c (1-(1-p e ) (1- p b p d )) 1-(1-p a )(1-p c (1- (1-p e )(1- p b p d )))

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14 Factoring A Naïve approach Reliability calculation costs too much. … p a p b p c p d p e + (1-p a )p b p c p d p e + p a (1-p b )p c p d p e + … papa pbpb pcpc pdpd pepe

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15 Factoring Concept Select one edge (e) R(G) = p e *R(Ge)+(1-p e )*R(G-e) Ge e G-e G Ge = graph obtained by contracting edge e in G G-e = graph obtained by deleting edge e in G When G − e is failed, any sequence of contractions and deletions results in a failed network Hence there is no need to factor G − e.

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16 a c e b d v1v1 v2v2 v3v3 v4v4 Minpaths of the system that 3 connected nodes are operating normally Quiz

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17 Review of Fault Tree

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18 Fault Trees Failure 2-out-of-3 S1S2S3 TMR k-out-of-n 0: Normal, 1: Fails OR gate AND gate i=k n ( ) F(t) i (1-F(t)) n-i n i F(t) = probability of the occurrence of failure event (function of F(t) is CDF) Pictorial representation of the combination of events that can cause the occurrence of an undesirable event (failure). Staring point (of tree) is the definition of a single undesirable event (failure). An event is reduced to a combination of low-level events by means of logic gates.

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19 Fault Tree Model & Reliability Block Model 2 processors (P)3 memory module (M) Failure P1M2M3 or and M1 P2 Fault Tree Reliability block diagram Reliability Block Model The structure that shows when the system is functioning. Fault Tree Model The structure that shows when the system has failed The output of the top event is a logic 1 0: Normal, 1: Fails

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20 Example of Fault Tree Failure and or S1S2S3 1-(1-F 2 (t))*(1-F 3 (t)) 0: Normal, 1: Fails F 1 (t)*(1-(1-F 2 (t))*(1-F 3 (t)))

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21 Fault Trees (Basic) 2 processors (P)3 memory module (M) Failure P1M2M3 or and M1P2 The system is operational if at least one processor and one memory module are operational. Fault Tree Reliability block diagram F(t) = ?? Fault Tree when there is no repeated component Failure distribution for the component is independent

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22 Fault Trees (Advance) Fault Tree when there is repeated component Failure distribution for the component is not independent Suppose that instead of all three memory modules being shared between the two processors, one of the memory modules (M3) is shared and the other two are private, one for each processor. What it reliability block diagram ? Failure and or M1M3 and or M2M3 P1P2 We have to use factoring technique !

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23 Factoring Failure and or M1M3 and or M2M3 P1P2 F(t) = F M3 (t)*F a (t) + (1-F M3 (t))*F b (t) Multiply the result for each case by the probability that case happens, then add the products. F(t)F(t) 0: Normal, 1: Fails Failure and or M1 P1 M2 P2 M3 has failed Fa(t)Fa(t) Failure and P1P2 M3 has not failed Fb(t)Fb(t)

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