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

Pattara LeelapruteComputer Engineering DepartmentKasetsart Universitypattara.l@ku.ac.th

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Review

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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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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 failedPe = Pr [edge e is operational] = reliability of e

Unnecessary to think about time (=availability)

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Fault Model

Situation of Network

a

c

e

b

d

v1

v2

v3

v4

pa =0.9pb =0.8pc =0.9pd =0.9pe =0.95

…

Pe = Pr [edge e is operational] = reliability of e

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Network Reliability

k-terminal reliabilityProbability that there exist operating paths

between every pair of nodes in K

Two terminal reliabilityProbability 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 nodesV = all nodes

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Minpaths Pathset

A set of components (edges) whose operation implies (guarantees) system operation

MinpathA minimal PathsetEx ． K={v1,v4}

v1 v4

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Mincuts Cutset

A set of components (edges) whose failure implies (guarantees) system failure

MincutA minimal CutsetEx ． K={v1,v4}

v1 v4

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a

c

e

b

d

v1

v2

v3

v4

Minpaths of the system that 3 successively connected nodes are operating normally

Quiz

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Computation of Reliability Complexity for two-terminal reliability and

all terminal reliability NP-hard (#P-complete)

AlgorithmsEfficient Algorithms for Restricted ClassesExponential time algorithm for general

networks

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Transformations and ReductionsR(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)

eu v

u(= v)

G: G’:

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Transformations and Reductions Parallel Reduction

G, G’

Multiplicative factor = 1 Series Reduction

G, G’

Multiplicative factor = 1

G: G’:p1 p2 p1 p2

p1

p21- (1- p1) (1- p2)

G: G’:

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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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An Example

Parallel Replacement

Series Replacement

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An Example

p1 p2

p1

p21- (1- p1) (1- p2)

p1 p2

papb

pcpd

pe

pa

pc

pe

pbpd pa

pc

pa

1-(1-pe)(1- pbpd)

pc(1-(1-pe)(1- pbpd))

1-(1-pa)(1-pc(1-(1-pe)(1- pbpd)))

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Algorithm to calculate K-terminal reliability

There exists an algorithm to calculate K-terminal reliability in polynomial time.

Factoring Sum of Disjoint Products (SDP)

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Factoring A Naïve approach

Reliability calculation costs too much.

…

papbpc pd pe + (1-pa)pbpc pd pe + pa(1-pb)pc pd pe + …

papb

pcpd

pe

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Factoring Concept

Select one edge (e)R(G) = pe*R(G•e)+(1-pe)*R(G-e)

G•e

e

G-e

G

G•e = graph obtained by contracting edge e in GG-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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Sum of Disjoint Products (SDP)

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Sum of Disjoint Products (SDP)

Approach implemented by using Boolean algebra Ex. Two terminal reliability between v1, v4

a

c

e

b

d

v1

v2

v3

v4

Minpath: ab, cd, ade, bce

Can be expressed with the following Boolean expression: = AB ∨ CD ∨ ADE ∨ BCE

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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)

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Sum of Disjoint Products (SDP)

Reliability = Pr [AB ∨ CD ∨ ADE ∨ BCE]

A￢ A

B

￢ B

￢ C ￢ CCC

DD

￢ D

￢ D

￢ E ￢ E￢ E ￢ EE E E E papb

pcpd

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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 termPr [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

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Sum of Disjoint Products (SDP)Reliability

A￢ A

B

￢ B

￢ C ￢ CCC

DD

￢ D

￢ D

￢ E ￢ E￢ E ￢ EE E E E

= 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

papb