John Carroll Cambridge University Engineering Department 'Proportional Fairness: Dynamics, Stability...

John CarrollCambridge University

Engineering Department

'Proportional Fairness: Dynamics, Stability and Pathology'

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Acknowledgements• (Nortel)

• Paul Kirkby• Sabesan Subramaniam. • Martin Biddiscombe• John Hudson• Radhakrishnan Kadengal• (Cambridge)

• Frank Kelly

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Background• Internet traffic needs differentiation of

services

• Premium traffic (perhaps different grades)

• ‘Guaranteed’ delivery at a price - different delays

• Best efforts - may be lost

• ‘Proportional Fairness’ is one scheme being studied for dynamic pricing to control access to network

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Outline• 1. Proportional Fairness - key concepts

• 2. Form of Network

• 3. Routes, Resources, Capacities

• 4. Theory of Small Changes• 5. Stability (frequency domain)

• 6. Stability (time domain)

• 7. Price Limited Proportional Fairness

• 8 Nil-Change and Max-Change Offers

• 9. Conclusions

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• Proportional Fairness - key concepts• 1. Each resource on network has price/bandwidth/

unit time specific to that resource. • This price varies with time. No limits.

• 2. All customers pay the price (Lp) for using the resource p

• 3. Resource p has a limited capacity (Cp).

• 4. Customers bandwidth-allocation along each route (covering many resources) is determined from the amount they are willing to pay for the route : ‘WtP’ or ‘bid’

• 5. All resource prices are continually adjusted so as to fill the capacity of all resources, given customers bid. JEC; JEC;

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• any network with pre-assigned routes of limited capacity.

• simple example of 4 node ring to illustrate algebra

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Node 1

Node 2 Node 4

Node 3

Network formed with 4 nodes servicing 16 routes

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Routes & pricesm11 m12m13m14m21m22m23m24 = m column vectorm31m32m33m34m41m42m43m44

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Willingness-to-pay

or bid price

x11 x12x13x14

x21 x22x23x24

= x x31 x32

x33x34

x41 x42x43x44

ordered

routes

Node 1

Node 2 Node 4

Node3

Resource 4 price L4capacity C4

Resource 3 price L3capacity C3

Resource 2 price L2

capacity C2

Resource 1 price L1

capacity C1

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• x11 m11 x12 m12 C1 L1x13 m13x14 m14

• x21 m21x22 m22 C2 L2x23 m23 =C =Lx24 m24

• x31 m31x32 m32 C3 L3x33 m33x34 m34

• x41 m41x42 m42 C4 L4x43 m43x44 m44

Capacities & price/bw. for resources

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• x11 started at tx12x13x14x21x22x23x24 started at t T x31

x32x33 started at t 2T x34 x41

x42 x43 started

at t T x44

Node 1

Node 2

Node 4

Node 3

T

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10 Routes filling Resource 1 - C1

2T

3T

Delay Operator• All delays in unit of hop-time T

(all hops taken of equal length)

• Delay operator z-1 f( t T) = z-1 f

• For single frequency then z-1 = exp( j T)

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If Capacity C1 Fully Used:-• C1= 1 x x11 + 1 x x12 + 1 x x13 + 1 x x14

+ 0 x x21 + 0 x x22 + 0 x x23+ z-3 x x24 + 0 x x31 + 0 x x32 + z-2 x x33+ z-2 x

x34 + 0 x x41 + z-1 x x42 + z-1 x x43+ z-1 x x44

• Similar calculations for capacities C2 , C3 and C4

• C = Scap x global matrix form

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Price/bandwidth for Routes• x11 uses L1

x12 uses L1 + L2x13 uses L1 + L2 + L3

x14 uses L1 + L2 + L3 + L4

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Node 1 as typical

Price/route That Must Be Offered

• x11 x (L1 ) = m11 x12 x (L1 + z-a L2) = m12 x13 x (L1 + z-a L2 + z-b L3) = m13 x14 x (L1 + z-a L2 + z-b L3+ z-c L4) = m14

• Yellow terms calculated directly (globally or locally) from previous capacity equations

• Delay operators depend on how prices become known to manager at node 1

• Additional smoothing can be introduced

• Similar calculations for nodes 2 , 3 & 4

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Offer/allocation = STRUCTURE* Price/resource

• m11 / x11 1 0 0 0m12 / x12 1 z-a 0 0 m13 / x13 1 z-a z-b 0 m14 / x14 1 z-a z-b z-c m21 / x21 0 1 0 0m22 / x22 0 1 z-a 0 L1 m23 / x23 0 1 z-a z-b L2m24 / x24 = z-c 1 z-a z-b L3m31 / x31 0 0 1 0 L4m32 / x32 0 0 1 z-a m33 / x33 z-b 0 1 z-a m34 / x34 z-b z-c 1 z-a m41 / x41 0 0 0 1m42 / x42 z-a 0 0 1 m43 / x43 z-a z-b 0 1m44 / x44 z-a z-b z-c 1

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Offer/allocation = STRUCTURE* Price/resource

• m11 / x11 1 0 0 0m12 / x12 1 Da 0 0 m13 / x13 1 Da Db 0 m14 / x14 1 Da Db dc m21 / x21 0 1 0 0m22 / x22 0 1 Da 0 L1 m23 / x23 0 1 Da Db L2m24 / x24 = Dc 1 Da Db L3m31 / x31 0 0 1 0 L4m32 / x32 0 0 1 Da m33 / x33 Db 0 1 Da m34 / x34 Db Dc 1 Da m41 / x41 0 0 0 1m42 / x42 Da 0 0 1 m43 / x43 Da Db 0 1

m44 / x44 Da Db dc 1

(m./x)

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MATLAB Notation

Sres L

Matrix Formulation of Network

• (m./x) = Sres L resources

• C = Scap x capacities

• z-1 = 1 in steady state when Scap = Sres

tr : incidence matrix

• Compact but non-linear because of 1/x

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• 4. Theory of Small Changes

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• m m + m bids• x x + x allocations of

bandwidth• C C + C resource capacities• L L + L resource prices• indicates small changes from

steady state. • Neglect x. x , x.m etc.

Matrix Formulation: Small Changes

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• Define: %m m. /m: fractional bid changes; %x x. /x : fractional alloc. changes;

• Xd as a diagonal matrix with diagonal elements x (allocations); steady state information.

• Md as a diagonal matrix with diagonal elements m (willingness to pay); steady state information.


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• Using the constraint on x it is possible to show that changes in bids m and resource prices L are linked

• ScapXd m = R L where• R = Scap XdMd

Xd Sres

• R must have inverse to find L• Det|R| 0 at all real JEC; JEC;

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• R matrix (Mnodex Mnode) encapsulates steady state values and information delays.

• Det|R| always has zeros at some complex frequency real j imag

• ‘Zeros’ give outputs with no inputs - transients

• If imag> 0 then transients grow• Instability

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• Small changes allow one to integratefrom approximate to ‘exact’ solutions.

• Allow integration from one exact solution to another with different bids

• Examine effects of ‘coalitions’ where customers combine in cartels.• Here we concentrate on stability of allocations in response to varying

bids


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Stability Criterion• Calculate a typical steady state;• Allow for small changes• Calculate y= log(abs(det|R|)) over a

grid of complex frequencies; note (det|R|)) periodic in real

• Seek minima in y ;• Check if minima deepen for increasing

imag> 0 in real j imag

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Example 1 reverse price propagation

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Node 1

Node 2

Node 4

Node 3

Propagation of Information about Price/bandwidth on resources

Resource 4: known to manager at node 1 from update at t= T

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Loge[abs(det R)] as a function of r T and i T(pricing information travels by reverse route)

increasingi T steps of 0.025,(decreasing depth of minima indicating stability)

0

1

2

3

4

r T

Linefori T = 0

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Example 2 forward price propagation

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Node 1

Node 2

Node 4

Node 3

Propagation of Information about Price/bandwidth on resources

Resource 4: known to manager at node 1 from update at t= T

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r T

Loge[abs(det R)] as a function of r T and i T(pricing information travels by completing ring)

increasingi T;steps of 0.0125,(initial increase in minima indicatinglocation of instability)

0

1

2

3

4

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Linefori T = 0

Stability Theorem• If feedback system is small signal

stable/unstable in one steady state, then it will be stable/unstable in neighbouring steady states.

• Margin of stability needs criteria -given from magnitude of imag

• Stability of ‘a calculation system’ (= 0) does not guarantee stability of dynamic system.






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Local Price DeterminationReverse Propagation of Price

• Local manager at each node controlling ingress of traffic to network

• Local manager controls local resource prices to match supply and demand ( bids)

• Local Manager informed of distant resource prices (delayed information)

• Has sufficient information to determine local resource prices and to determine a locally proportionally fair allocation

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Example :reverse price propagation with local price control

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4 8 12 16

5

1

28 initial start

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Normalised offers dynamic allocations _____o

Timestep

Group 1 Group 2 Group 3 Group 4

Route no.

29 - step change

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1

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Route no.

31

4 8 12 16

5

1

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Route no.

34

4 8 12 16

5

1

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Route no.

36

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1

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Steady state allocation

Route no.

System stabilises for even quite large changes in 4 - 12 time steps (1 to 3 circuits)

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• 7. Price Limited Proportional Fairness

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‘Pathology’ of Proportional fairness

• All may not seem fair in a proportionally fair system

• One set of bids m gives bandwidth allocations x then k m any another set where k is the same for all routes gives same allocations!

• even if k =0.001 or 10 there is still no change in allocation to any one route

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• Price Limited Proportional Fairness• 1. Each resource on network has price/bandwidth/

unit time specific to that resource. • This price varies with time. Not allowed to fall

below a specified level at each resource.

• 2. All customers pay the price (Lp) for using the resource p

• 3. Resource p has a limited capacity (Cp).

• 4. Customers bandwidth-allocation along each route (covering many resources) is determined from the amount they are willing to pay for the route : ‘WtP’ or ‘bid’

• 5. Capacity of all resources is only filled if customers pay sufficient. JEC; JEC;

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Dynamic Example:reverse price propagation with local price controland Price-Limited Proportional Fairness(PLPF)

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4 8 12 16

5

1

28 initial start

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S P F OfferPLP F

Timestep

Group 1 Group 2 Group 3 Group 4

Route no.

30

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5

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Better PLPFcontrol

S P F OfferPLP F

Route no.

32

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5

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S P F OfferPLP F

Route no.

34

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5

1

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S P F OfferPLP F

Route no.

36

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Steady state SPF

Normalised Offer

Route no.

Steady state PLPF

System still stabilises for even quite large changes in 4 - 12 time steps (1 to 3 circuits)

LPF reduces pathological allocations and speeds stabilisation when offers are low.

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• 7. Limited proportional fairness

• 8 Nil-Change and Max-Change Offers

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• Make an arbitrary offer given as a fractional change %m

• The algebra enables definition of matrix P such that change of allocation in response to change of bids is %x[ I - P ] %m [ PP=P ]

• write %mnilP %m• Associated %xfor %mnil is %x

Nil-change bid• remaining part of bid

%mmax gives %x%mmax

Max-change bid

Matrix formulation: Nil-Change+Max-Change

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• Max-change and nil-change still useful even when changes are relatively large.

• All changes of bid can be split into nil-change + max-change.

Nil-Change+Max-Change

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Significance of Nil-Change and Max-Change bids

• Nil-increase offers can create cartels where customers gang up on network operator - or vice versa : unfortunate collective changes of bids with no change in allocations to customers.

• Max-change offers should improve customer satisfaction: customers obtain what they expect.

• Statistical analyses needed to control nil-change offers?

• Route diversification should increase ratio of max-change components to nil-increase components.

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Example of projecting out Nil-increase and Max-Change offers

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An aside: normalisation• A standard set of equilibrium bids is useful for

normalising results: all normalised flows = 1 all normalised offer prices = 1.

• For our network the standard flows are all unit value and the standard offers are[ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4]:- offer/route proportional to number of hops.

• To normalise, offer prices are divided by the standard value.

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0

Total -change integrated upfrom series of small changes

4 8 12 16

1

2

3

Route number

Total normalised offer (offer price dashed - allocation solid)

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frac

tiona

l cha

nge

4 8 12 16Route number

0

1

2

-1

Nil-increase component of fractional changein normalised offer (offer price dashed - allocation solid)

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4 8 12 16Route number

frac

tiona

l cha

nge

0

1

2

-1

Max-change component of fractional changein normalised offer; here the fractional change in allocation coincided with fractional change in offer.

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9. Conclusions

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• Matrix methods organises solutions for large scale networks both locally and globally

• Small changes allows integration over large changes• Small signal frequency domain stability considers a single

determinant of the scale of the resource numbers - not on the scale of the numbers of routes.

• Small signal frequency domain stability re-assures one about large signal stabiilty

• Stability and instability have both been demonstrated as predicted.

• Price Limited Proportional Fairness can prevent pathological allocations of unwanted bandwidth.

• Small changes show pathology: (a) nil-increase offers - dangers of cartels (b) max-change offers - optimise customer satisfaction

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John Carroll Cambridge University Engineering Department 'Proportional Fairness: Dynamics, Stability...

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