1. Fault-Tolerant Computing Systems #7 Network Reliability 2 & Sum of Disjoint Products
Pattara Leelaprute
Computer Engineering Department
Kasetsart University

2. Review

3. 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

4. Network Reliability Problems Network Model
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
Pe = Pr [edge e is operational] = reliability of e
Unnecessary to think about time (=availability)

5. Fault Model Situation of Network
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 v1 v4 c d v3
Situation of Network …

6. Network Reliability k-terminal reliability Two terminal reliability
K = set of nodes
V = all nodes
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)

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

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

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

10. 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

11. 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)

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

13. 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.

14. An Example
Series Replacement
Parallel Replacement

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

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

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

18. Factoring Concept G 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

19. Sum of Disjoint Products (SDP)

20. 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

21. 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)

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

23. 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

24. 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