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´ ozsef J. B´ ır´ o, 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´ osok k¨ or´ 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 ( s 1 , d 1 ) = (1 , 3) ( s 2 , d 2 ) = (2 , 3) A graph G ( V, E ) Edge capacities u Source-destination pairs ( s k , d k ) : k ∈ K – p. 3

How does geometry come into the picture? Given a network G u – p. 4

How does geometry come into the picture? Given a network G u The flow polytope M ( G u ) describes all the routable path-flows – p. 4

Polytopes intersection of half- spaces – p. 5

Polytopes intersection of half- convex combination of spaces points – p. 5

The throughput polytope G u – p. 6

The throughput polytope G u M ( G u ) – p. 6

The throughput polytope θ 1 = f 1 + f 2 θ 2 = f 3 G u T ( G u ) M ( G u ) – p. 6

Properties of T ( G u ) “The set of traffic matrices realizable in G u ” – p. 7

Properties of T ( G u ) “The set of traffic matrices realizable in G u ” Polytope – p. 7

Properties of T ( G u ) “The set of traffic matrices realizable in G u ” Polytope Full-dimensional – p. 7

Properties of T ( G u ) “The set of traffic matrices realizable in G u ” Polytope Full-dimensional Down-monotone – p. 7

Another network ( s 1 , d 1 ) = (1 , 5) ( s 2 , d 2 ) = (2 , 5) ( s 3 , d 3 ) = (3 , 5) T ( G u ) G u – p. 8

Minimum cuts (in the Ford-Fulkerson-sense) ( s 2 , d 2 ) = (2 , 5) maximum flow = minimum capacity cut θ 2 ≤ 1 – p. 9

Minimum cuts (in the multicommodity-sense) separating edges of minimal capacity θ 1 + θ 2 ≤ 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 one 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 off without hurting someone else who is already poorer – a unique max-min fair allocation exists over T ( G u ) – only depends on G u – independent of any routing whatsoever θ 0 = [ 1 2 , 1 2 ,1 ] – p. 14

Bottlenecks (in the traditional sense) A bottleneck edge (of some user k ) is – filled to capacity – θ k 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 – θ k is maximal θ 0 = [ 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 θ 0 = [ 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 ( G u ) ? – ray-shooting – p. 18

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