el736 communications networks ii: design and algorithms
DESCRIPTION
EL736 Communications Networks II: Design and Algorithms. Class9: Fair Networks Yong Liu 11/14/2007. Outline. Fair sharing of network resource Max-min Fairness Proportional Fairness Extension. Fair Networks. Elastic Users: demand volume NOT fixed - PowerPoint PPT PresentationTRANSCRIPT
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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: definition
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 properties
Generalized notions of fairness [Mo & Walrand, 2000] -proportional fairness: A feasible rate vector x
isfair 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
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
Procedure1. 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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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 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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−∑i
iii x
xyw