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Fair Network Resource Allocation: Max-Min and Proportional Fairness Designs

This document outlines methodologies for fair sharing of network resources, focusing on Max-min and Proportional Fairness. It discusses the concepts of fair allocation amidst elastic users with variable demands and the competition for resources like bandwidth and budget. Key algorithms for achieving Max-min fairness are explored, detailing extensions to handle uncapacitated scenarios and multiple paths for demands. Additionally, it addresses rate control and bandwidth reservation applications, providing insight into unique formulated propositions and decentralized algorithms introduced in the context of network resource management.

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Fair Network Resource Allocation: Max-Min and Proportional Fairness Designs

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  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

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