L14. Fair networks and topology design

D. Moltchanov, TUT, Spring 2008

D. Moltchanov, TUT, Spring 2015

Fair networks

FN: why fairness?So far we considered constant demands, e.g. 10Mbps

Is this realistic for modern networks? Yes and noDepends on the type of protocol/application/network

Constant demandsReal-time applications, e.g. voice videoRunning over UDP or RTP/UDP protocols

Elastic greedy demandsOccupies whatever you offerPossibly within some upper boundsTCP-like protocolsRate-adaptive real-time applications

Real networks: traffic always has some elasticity

FN: demands are variables!General problem here

How much to assign to demandsSuch that capacity of links is not exceeded!

We have no actual values of demands availableA user (demand) is supposed to be happy with anything we offerA user is supposed to take anything we offer

Demands are also variable, not only

We need to allocate in a fair way…

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FN: MAX-MIN fairness criterionFairly straightforward approach

Assign a certain possible minimum to all demandsIf constraints are not satisfied – problem is infeasibleIf satisfied – try to increase allocation…

Max-min fairness criterionAs fair as possible for all usersIrrespective of path lengthsNetwork throughput could be a problem (you’ll see why soon)

Implementation: water filling algorithmAssign the same maximum possible for all demandsThis assures that the minimal assignment is maximizedIncrease assignment for other flows

Relatively easy when only single paths are available

FN: MAX-MIN exampleConsider three nodes scenario

Linear topology with three demandsNo demand values: greedy elastic trafficLink capacities: Interfering demands

Analyzing the problemAll demands have exactly one pathThe same maximum for all

assignment for d=3 is obviously maximizedand… we are finished, no more improvement for demandsif no: continue improving for d=1 and d=2 (with some step)until there are no demands lefte.g. let after 0.5 left for d=2

1.5, 1,2ec e

11 21 31 0.75x x x

11 21 31(1), (2), (1,2)P P P

2 2c 11 21 31 0.75x x x

11 31 210.75, 0.75 0.5 1.25x x x

FN: MAX-MIN exampleWhat is the problem with max-min fairness (MMF)?

Fair for users! Is this good?Why should I get less if my connection shorter than yours?e.g. accessing www.tampere.fi from Helsinki and US…Not practical example but you see the point

Another problem?What is about overall network throughput?

What is the maximum one? usageCan we achieve it? Yes!

Good for operator, unfair for users

Is there a kind of balance?

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2.25 / 3 0.75%U 3T

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FN: proportional fairnessMax-min fairness is always like this

Good for end usersBad for operators

Recall TCP behaviorRate is inverse proportional to route trip time (RTT)Mathis expression for TCP rate

where L – packet size, p – packet loss probabilitythe more links we have the higher the RTT and slower the rateTCP demonstrate good results is so-called “fair protocol”Different principle, not max-min!!!

Compromise: so-called proportional fairness criterion

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LR p RTT

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FN: proportional fairnessProportional fairness (PF)

Maximize logarithms of allocations

In our particular case

Why logarithmic function?Very small when Does not allow for zero allocations

Quickly increasesAllows for find precise balance

Slowly increases when for large Prevent from allocating all to one or few paths

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0 2 4 6 8 104

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4f(x) = log x

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FN: proportional fairnessSolution for PF in our case

Can be found analyticallySimple topology, single paths, etc.

Our task is to maximize

due to symmetricity

leading to

Solution: second derivative testGet critical points If then a is maximumSolution: throughput

0 0.25 0.5 0.75 1 1.25 1.510

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f(x) =2 ln x( ) ln 1.5 x( )

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1x

11 12 311, 0.5x x x 2.5T 0.833U

FN: proportional fairnessProportional fairness in general

Better for operators! … and worse for users?Solution is “more clear” than that for MMF

Can be extended to add weights to allocated demands

parameters add additional flexibility

Generalized notion of fairnessAllocation vector is if for any other feasible vector

when we have proportional fairnesswhen we have proportional fairness

PF: convex programming problem, could be linearized

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fair ( )

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FN: MAX-MIN exampleWhy MAX-MIN could be complex!

Nothing to minimize/maximizeSolved in terms of lexicographical sorting of allocation vectors

Denoting: we have the following definitions

MAX-MIN fair allocation

MAX-MIN fair allocation

A feasible allocation vector is max-min fair if no allocation can be increased without simultaneously decreasing some for which we have .

X

A feasible rate allocation vector is an optimal solution to the max-min problem if for every feasible rate vector with , for some demand , there exist a demand such that and .

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FN: MAX-MIN exampleProblem when multiple paths…

or

Filling algorithm: which paths to fill?Maximize minimum for allFirst paths set:Second paths set:

Any improvements?First set of paths: for demand d=1

Second set of paths: no improvement possible…We should avoid such situations!More details in Pioro, Medhi “Flow and capacity design…”

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11 21(1,3), (2,3,4)P P

(1) (2) (2)11 21( , ) (1,1)x x x

11 11 12 12 21 211, (1) 1, (1,3) 1, (1,4)x P x P x P

Topology design

TD: costs for deploying a networkUncapacitated NDP: we took into account

Capacity-dependent cost of links, recall objective function

where is the cost of capacity unit on link ewhat is about deployment costs?often this is the actual price of a link!

1

E

e ee

F y

e

TD: costs for deploying a networkThe objective function for this problem is

where indicated whether a link is installed or notparameters shows the installation cost

Problem definition:

minimize subject to

and constraint: where is large constant

Note: the latter is needed to force when e is not installed

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TD: exampleOur four nodes example

Parameters:

The optimal topology is a treeShown by red linksLinks: 2, 3, and 4,

Installation cost:

Capacity costs: using we get

Topology design problemsLinear programming problemsEasy to solveAllows extensions: e.g. modular links

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