Communications Networks II: Design and Algorithms

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Routing, Flow, and Capacity Design in
Communication and Computer NetworksChapter 8
Fair Networks
Slides by Yong Liu1, Deep Medhi2, and Michal
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 resourcebandwidth
• uncapacitated design resourcebudget
• 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 (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?

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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
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

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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 ABBC1, CA2
B
demand 4 2/3
demand 1,2,3 1/3
C
A
demand 54/3
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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