Fault-Tolerant Computing Systems #7 Network Reliability 2 & Sum of Disjoint Products
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Transcript of Fault-Tolerant Computing Systems #7 Network Reliability 2 & Sum of Disjoint Products
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Fault-Tolerant Computing Systems#7 Network Reliability 2 & Sum of Disjoint Products
Pattara LeelapruteComputer Engineering DepartmentKasetsart [email protected]
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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