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Routing, Flow, and Capacity Design in Communication and Computer Networks Chapter 8: Fair Networks
Slides by Yong Liu1, Deep Medhi2, and Michał Pióro3 1Polytechnic University, New York, USA 2University of Missouri-Kansas City, USA 3Warsaw University of Technology, Poland & Lund University, Sweden October 2007
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Outline Fair sharing of network resource Max-min Fairness
Proportional Fairness Extension
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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
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Max-Min Fairness: definiation
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?
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Other Fairness Measures
Proportional fairness [Kelly, Maulloo & Tan, ’98] A feasible rate vector x is 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
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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.
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Max-min Fairness Example
Session 3 Session 2 Session 1 C=1 C=1 Session 0 Max-min fair flow allocation sessions 0,1,2: flow rate of 1/3 session 3: flow rate of 2/3
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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
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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.
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Max-min Fair Algorithm
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Max-min Fair Example link rate: AB=BC=1, CA=2 B demand 4 =2/3
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Extended MMF lower and upper bound on demands weighted demand rate
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Extended MMF: algorithm
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Deal with Upper Bound Add one auxiliary virtual link with link capacity wdHd for each demand with upper bound Hd
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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
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Uncapacitated Problem
Max-min fair sharing of budget Formulation
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Uncapacitated Problem
max-min allocation all demands have the same rate each demand takes the shortest path proof?
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Proportional Fairness
Proportional Fairness [Kelly, Maulloo & Tan, ’98] A feasible rate vector x is proportionally fair if for every other feasible rate vector y formulation
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Linear Approximation of PF
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Extended PF Formulation
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Uncapacitated PF Design
maximize network revenue minus investment
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