Fairness in Capacitated Networks: a Polyhedral Approach G abor R - - PowerPoint PPT Presentation

Fairness in Capacitated Networks: a Polyhedral Approach

G´ abor R´ etv´ ari, J´

• zsef J. B´

ır´

• , Tibor Cinkler

{retvari,biro,cinkler}@tmit.bme.hu

High Speed Networks Laboratory Department of Telecommunications and Media Informatics Budapest University of Technology and Economics H-1117, Magyar Tud´

• sok k¨
• r´

utja 2., Budapest, HUNGARY

– p. 1

Agenda

Model: the Geometry of Networking Application: fair throughput allocations in capacitated networks

– p. 2

A network

A graph G(V, E)

– p. 3

A network

A graph G(V, E) Edge capacities u

– p. 3

A network

(s1, d1) = (1, 3) (s2, d2) = (2, 3)

A graph G(V, E) Edge capacities u Source-destination pairs (sk, dk) : k ∈ K

– p. 3

How does geometry come into the picture?

Given a network Gu

– p. 4

How does geometry come into the picture?

Given a network Gu The flow polytope M(Gu) describes all the routable path-flows

– p. 4

Polytopes

intersection

• f

half- spaces

– p. 5

Polytopes

intersection

• f

half- spaces convex combination of points

– p. 5

The throughput polytope

Gu

– p. 6

The throughput polytope

Gu M(Gu)

– p. 6

The throughput polytope

Gu M(Gu) θ1 = f1 + f2 θ2 = f3 T(Gu)

– p. 6

Properties of T(Gu)

“The set of traffic matrices realizable in Gu”

– p. 7

Properties of T(Gu)

“The set of traffic matrices realizable in Gu” Polytope

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Properties of T(Gu)

“The set of traffic matrices realizable in Gu” Polytope Full-dimensional

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Properties of T(Gu)

“The set of traffic matrices realizable in Gu” Polytope Full-dimensional Down-monotone

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Another network

(s1, d1) = (1, 5) (s2, d2) = (2, 5) (s3, d3) = (3, 5) Gu T(Gu)

– p. 8

Minimum cuts (in the Ford-Fulkerson-sense)

(s2, d2) = (2, 5)

maximum flow = minimum capacity cut

θ2 ≤ 1

– p. 9

Minimum cuts (in the multicommodity-sense)

separating edges of minimal capacity

θ1 + θ2 ≤ 1

– p. 10

Fairness in capacitated networks

An allocation of user throughputs that is – realizable – efficient – rightful Challenge: solve this problem without having to fix the paths

θ = [1

2,1 2,1]

– p. 11

Fairness in capacitated networks

An allocation of user throughputs that is – realizable – efficient – rightful Challenge: solve this problem without having to fix the paths

θ = [1

3,1 3,1 3]

– p. 11

Fairness in capacitated networks

An allocation of user throughputs that is – realizable – efficient – rightful Challenge: solve this problem without having to fix the paths

θ = [2

5,2 5,2 5]

– p. 11

Efficient allocations (Non-dominatedness)

Definition: at least

• ne user is blocked

Location: at the boundary Problem: too wide a definition

– p. 12

Efficient allocations (Pareto-efficiency)

Definition: no way to make any person better off without hurting anybody else Location: at certain faces Problem: allows for dictatorship

– p. 13

Max-min fairness

Definition: no way to make anybody better

• ff without hurting

someone else who is already poorer – a unique max-min fair allocation exists over

T(Gu)

– only depends on Gu – independent

• f

any routing whatsoever

θ0 = [1

2,1 2,1]

– p. 14

Bottlenecks (in the traditional sense)

A bottleneck edge (of some user k) is – filled to capacity – θk is maximal at the edge Water-filling algorithm

θ = [2

5,2 5,2 5]

– p. 15

Generic bottlenecks

Geometrically: bottlenecks ≡ valid inequalities Graph-theoretically: bottlenecks ≡ separating edge sets – filled to capacity by any routing – θk is maximal

θ0 = [1

2,1 2,1]

– p. 16

Water-filling

Find at least one bottleneck in each iteration – start along the ray

θ = [1, 1, 1]

– proceed until blocked – continue along non-blocked users

θ0 = [1

2,1 2,1]

– p. 17

Conclusions

Geometry of Networking – flow-theoretic reasoning – geometric argumentation Network fairness: a side-product – routing-independent max-min fair allocation – exists and unique – a bottleneck argumentation (in fact, 2 ones) – water-filling How to compute T(Gu)? – ray-shooting

– p. 18

Limitations

1 10 100 (K) 9 8 7 6 5 4 3 2 1 Europe28(avg) Germany(avg) NSF(avg) max

– p. 19

Further applications

State aggregation for inter-domain traffic engineering – hides topological information – reveals just enough detail Admission control Routing Network decomposition

– p. 20