EL736 Communications Networks II: Design and Algorithms

1. EL736 Communications Networks II: Design and Algorithms Class9: Fair Networks Yong Liu 11/14/2007

2. Outline • Fair sharing of network resource • Max-min Fairness • Proportional Fairness • Extension

3. Fair Networks • Elastic Users: • demand volume NOT fixed • greedy users: use up resource if any, e.g. TCP • competition resolution? • Fairness: how to allocate available resource among network users. • capacitated design: resource=bandwidth • uncapacitated design: resource=budget • Applications • rate control • bandwidth reservation • link dimensioning

4. Max-Min Fairness: definition • Lexicographical Comparison • a n-vector x=(x1,x2, …,xn) sorted in non-decreasing order (x1≤x2 ≤ …≤ xn) is lexicographically greater than another n-vector y=(y1,y2, …,yn) sorted in non-decreasing order if an index k, 0 ≤k ≤n exists, such that xi =yi, for i=1,2,…,k-1 and xk >yk • (2,4,5) >L (2,3,100) • Max-min Fairness: an allocation is max-min fair if its lexicographically greater than any feasible allocation • Uniqueness?

5. Other Fairness Measures Proportional fairness[Kelly, Maulloo & Tan, ’98] • A feasible rate vector xis proportionally fair if for every other feasible rate vector y • Proposed decentralized algorithm, proved properties Generalized notions of fairness [Mo & Walrand, 2000] • -proportional fairness: A feasible rate vector x is fair if for any other feasible rate vector y • Special cases: : proportional fairness : max-min fairness

6. Capacitated Max-Min Flow Allocation • Fixed single path for each demand • Proposition: a flow allocation is max-min fair if for each demand d there exists at least one bottle-neck, and at least on one of its bottle-necks, demand d has the highest rate among all demands sharing that bottle-neck link.

7. Max-min Fairness Example • Max-min fair flow allocation • sessions 0,1,2: flow rate of 1/3 • session 3: flow rate of 2/3 Session 3 Session 2 Session 1 C=1 C=1 Session 0

8. Max-Min Fairness: other definitions • Definition1: A feasible rate vector is max-min fair if no rate can be increased without decreasing some s.t. • Definition2: A feasible rate vector is an optimal solution to the MaxMin problem iff for every feasible rate vector with , for some user i, then there exists a user k such that and

9. How to Find Max-min Fair Allocation? • Idea: equal share as long as possible • Procedure • start with 0 rate for all demand • increase rate at the same speed for all demands, until some link saturated • remove saturated links, and demands using those links • go back to step 2 until no demand left.

10. Max-min Fair Algorithm

11. Max-min Fair Example link rate: AB=BC=1, CA=2 B demand 4 =2/3 demand 1,2,3 =1/3 A C demand 5=4/3

12. Extended MMF • lower and upper bound on demands • weighted demand rate

13. Extended MMF: algorithm

14. Deal with Upper Bound • Add one auxiliary virtual link with link capacity wdHd for each demand with upper bound Hd

15. MMF with Flexible Paths • one demand can take multiple paths • max-min over aggregate rate for each demand • potentially more fair than single-path only • more difficult to solve

16. Uncapacitated Problem • Max-min fair sharing of budget • Formulation

17. Uncapacitated Problem • max-min allocation • all demands have the same rate • each demand takes the shortest path • proof?

18. Proportional Fairness • Proportional Fairness [Kelly, Maulloo & Tan, ’98] • A feasible rate vector xis proportionally fair if for every other feasible rate vector y • formulation

19. Linear Approximation of PF

20. Extended PF Formulation

21. Uncapacitated PF Design • maximize network revenue minus investment