Fault-Tolerant Computing Systems #7 Network Reliability 2 & Sum of Disjoint Products

Fault-Tolerant Computing Systems#7Network Reliability 2 & Sum of Disjoint Products Pattara Leelaprute Computer Engineering Department Kasetsart University pattara.l@ku.ac.th

Review

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

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

Fault Model Pe = Pr [edge e is operational] = reliability of e v2 pa =0.9 pb =0.8 pc =0.9 pd =0.9 pe =0.95 a b e v4 v1 c d v3 Situation of Network …

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

Minpaths • Pathset • A set of components (edges) whose operation implies (guarantees) system operation • Minpath • A minimal Pathset • Ex．K={v1,v4} v4 v1

Mincuts • Cutset • A set of components (edges) whose failure implies (guarantees) system failure • Mincut • A minimal Cutset • Ex．K={v1,v4} v4 v1

Quiz v2 Minpaths of the system that 3 successively connected nodes are operating normally a b e v4 v1 c d v3

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

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, G•e) • Multiplicative factor = pe • When e is mandatory (mandatory = an edge that appears in every minpath) u v G: G’: e u (= v)

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

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.

An Example Series Replacement Parallel Replacement

An Example 1-(1-pe)(1- pbpd) pb pbpd pa pa pa pa pe pe pc pc pd pc pc(1-(1-pe) (1- pbpd)) p1 p2 p1 p2 p1 1- (1- p1) (1- p2) 1-(1-pa)(1-pc(1-(1-pe)(1- pbpd))) p2

Algorithm to calculate K-terminal reliability There exists an algorithm to calculate K-terminal reliability in polynomial time. • Factoring • Sum of Disjoint Products (SDP)

pb pa pe pd pc Factoring • A Naïve approach • Reliability calculation costs too much. … papbpc pd pe +(1-pa)pbpc pd pe + pa(1-pb)pc pd pe + …

Factoring • Concept • Select one edge (e) • R(G) = pe*R(G•e)+(1-pe)*R(G-e) G•e = graph obtained by contracting edge e in G G-e = graph obtained by deleting edge e in G G•e • 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. G e G-e

Sum of Disjoint Products (SDP)

Sum of Disjoint Products (SDP) • Approach implemented by using Boolean algebra • Ex. Two terminal reliability between v1, v4 Minpath: ab, cd, ade, bce v2 a b e v1 v4 Can be expressed with the following Boolean expression:  = AB ∨ CD ∨ ADE ∨ BCE c d v3

Sum of Disjoint Products (SDP) • Reliability R(G) = Pr [AB ∨ CD ∨ ADE ∨ BCE = 1] • Probability for each path which operates correctly can be simply calculated as follows: Pr[AB]=papb, Pr[CD]=pcpd, ... • However, R(G) can not be directly calculated when there exists Pr of the paths which are not disjoint event (exclusive)

Sum of Disjoint Products (SDP) Reliability = Pr [AB ∨ CD ∨ ADE ∨ BCE] ￢A A ￢C C C ￢C papb ￢E E E ￢E ￢E E E ￢E ￢D B D D ￢B ￢D pcpd

Sum of Disjoint Products (SDP) • SDP Algorithm • Transform the Boolean expression so that each product term is exclusive for each other. AB ∨ CD ∨ ADE ∨ BCE = AB ∨ (￢A )CD ∨ (A￢B)CD ∨ (￢B)(￢C)ADE ∨(￢A)(￢D)BCE = AB ∨ (￢A ∨ A￢B)CD ∨ (￢B)(￢C)ADE ∨(￢A)(￢D)BCE • Reliability = Sum of probability (Pr) of each product term Pr [AB ∨ (￢A ∨ A￢B)CD ∨ (￢B)(￢C)ADE ∨(￢A)(￢D)BCE] = papb + ((1-pa) + (pa(1-pb))pcpd + (1-pb)(1-pc)papdpe + (1-pa)(1-pd)pbpcpe

Sum of Disjoint Products (SDP) Reliability = Pr [AB ∨ CD ∨ ADE ∨ BCE] = Pr [AB ∨ (￢A ∨ A￢B)CD ∨ (￢B)(￢C)ADE ∨(￢A)(￢D)BCE] = papb + ((1-pa)+(pa(1-pb))pcpd + (1-pb)(1-pc)papdpe+(1-pa)(1-pd)pbpcpe ￢A A ￢C C C ￢C papb ￢E E E ￢E ￢E E E ￢E ￢D B D D ￢B ￢D

Fault-Tolerant Computing Systems #7 Network Reliability 2 & Sum of Disjoint Products