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Slides byYong Liu1, Deep Medhi2, and Michał Pióro3

1Polytechnic University, New York, USA2University of Missouri-Kansas City, USA

3Warsaw University of Technology, Poland & Lund University, Sweden

October 2007

Routing, Flow, and Capacity Design in Communication and Computer

NetworksChapter 8:

Fair Networks

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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 MeasuresProportional 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 propertiesGeneralized 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

0)(

i

iii x

xyw

0)(

i

iii x

xyw

),( p

1

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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

Max-min fair flow allocation sessions 0,1,2: flow rate of 1/3 session 3: flow rate of 2/3

C=1 C=1

Session 1Session 2 Session 3

Session 0

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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

1. start with 0 rate for all demand2. increase rate at the same speed for all

demands, until some link saturated3. remove saturated links, and demands using

those links4. 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

B

CA

link rate: AB=BC=1, CA=2

demand 1,2,3 =1/3 demand 4 =2/3

demand 5=4/3

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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

0)(

i

iii x

xyw

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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