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EL736 Communications Networks II: Design and Algorithms

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EL736 Communications Networks II: Design and Algorithms Class9: Fair Networks Yong Liu 11/14/2007 Outline Fair sharing of network resource Max-min Fairness ... – PowerPoint PPT presentation

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Title: 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 resourcebandwidth
• uncapacitated design resourcebudget
• Applications
• rate control
• bandwidth reservation

4
Max-Min Fairness definition
• Lexicographical Comparison
• a n-vector x(x1,x2, ,xn) sorted in
non-decreasing order (x1x2 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
i1,2,,k-1 and xk gtyk
• (2,4,5) gtL (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 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

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
Session 3
Session 2
Session 1
C1
C1
Session 0
• Max-min fair flow allocation
• sessions 0,1,2 flow rate of 1/3
• session 3 flow rate of 2/3

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
• go back to step 2 until no demand left.

10
Max-min Fair Algorithm
11
Max-min Fair Example
link rate ABBC1, CA2
B
demand 4 2/3
demand 1,2,3 1/3
C
A
demand 54/3
12
Extended MMF
• lower and upper bound on demands
• weighted demand rate

13
Extended MMF algorithm
14
Deal with Upper Bound
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 x is 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