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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
  • link dimensioning

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
    links
  • 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
  • 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 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
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