dualtiy model of tcp/aqm - caltech · pdf filesteven low cs & ee, caltech...

Copyright, 1996 © Dale Carnegie & Associates, Inc.

Dualtiy Model of TCP/AQM

Steven Low

CS & EE, Caltechnetlab.caltech.edu

2002

Acknowledgments

S. Athuraliya, D. Lapsley, V. Li, Q. Yin (UMelb)S. Adlakha (UCLA), D. Choe(Postech/Caltech), J. Doyle (Caltech), K. Kim (SNU/Caltech), L. Li (Caltech), F.Paganini (UCLA), J. Wang (Caltech)L. Peterson, L. Wang (Princeton)

Network protocols.

HTTP

TCP

IP

Routers

Files

packetspacketspacketspacketspacketspackets

from J. Doyle

Web servers

web traffic

Is streamed out on the net.

Creating internet traffic

Webclient

from J. Doyle

Congestion Control

Heavy tail Mice-elephants

Elephant

Internet

Mice

Congestion control

efficient & fair sharing small delay

queueing + propagation

CDN

TCP

xi(t)

TCP & AQM

xi(t)

pl(t)

Example congestion measure pl(t)Loss (Reno)Queueing delay (Vegas)

Outline

Protocol (Reno, Vegas, RED, REM/PI…)

EquilibriumPerformance

Throughput, loss, delayFairnessUtility

DynamicsLocal stabilityCost of stabilization

))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+

TCP & AQM

xi(t)

pl(t)

Duality theory equilibrium Source rates xi(t) are primal variablesCongestion measures pl(t) are dual variablesFlow control is optimization process

TCP & AQM

xi(t)

pl(t)

Control theory stabilityInternet as a feedback systemDistributed & delayed

Outline

TCP/AQM Reno/RED

EquilibriumDuality model

Stability & optimal controlLinearized model

A scalable control

TCP & AQM

xi(t)

pl(t)

TCP: RenoVegas

AQM:DropTailREDREM/PIAVQ

TCP/AQMTahoe (Jacobson 1988)

Slow StartCongestion AvoidanceFast Retransmit

Reno (Jacobson 1990)Fast Recovery

Vegas (Brakmo & Peterson 1994)New Congestion Avoidance

RED (Floyd & Jacobson 1993)REM/PI (Athuraliya et al 2000, Hollot et al 2001)AVQ (Kunniyur & Srikant 2001)

Model structure

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

Multi-link multi-source network

x y

q p

from F. Paganini

TCP Reno (Jacobson 1990)

SStime

window

CA

SS: Slow StartCA: Congestion Avoidance Fast retransmission/fast recovery

TCP Vegas (Brakmo & Peterson 1994)

SStime

window

CA

Converges, no retransmission… provided buffer is large enough

queue size

for every RTT

{ if W/RTTmin – W/RTT < α then W ++

if W/RTTmin – W/RTT > α then W -- }

for every loss

W := W/2

Vegas model

pl(t+1) = [pl(t) + yl (t)/cl - 1]+Gl:

( )

<+=+ iiiii

ii dtqtxD

txtx α)()( if 1 )(1 2

( ) else )(1 txtx ss =+

Fi:

( )

>−=+ iiiii

ii dtqtxD

txtx α)()( if 1 )(1 2

Vegas model

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

x y

q p

+

−+= 1)()(

l

lll c

tytpG

( ))()(sgn ))((

1 )( 2 tqtxdtqd

txF iiiiii

ii −+

+= α

Overview





))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+

Outline

TCP/AQMReno/RED

EquilibriumDuality model

StabilityLinearized model

A scalable control

Flow control

Interaction of source rates xs(t) and congestion measures pl(t)Duality theory

They are primal and dual variables Flow control is optimization process

Example congestion measureLoss (Reno)Queueing delay (Vegas)

Model

c1 c2

NetworkLinks l of capacities cl

Sources iL(s) - links used by source iUi(xi) - utility if source rate = xi

x1

x2x3

121 cxx ≤+ 231 cxx ≤+

Primal problem

Llcy

xU

ll

iii

xi

∈∀≤

∑≥

, subject to

)( max0

AssumptionsStrictly concave increasing Ui

Unique optimal rates xi existDirect solution impractical

Related WorkFormulation

Kelly 1997Penalty function approach

Kelly, Maulloo & Tan 1998Kunniyur & Srikant 2000

Duality approachLow & Lapsley 1999Athuraliya & Low 2000

ExtensionsMo & Walrand 2000La & Anantharam 2000

DynamicsJohari & Tan 2000, Massoulie 2000, Vinnicombe 2000, …Hollot, Misra, Towsley & Gong 2001Paganini 2000, Paganini, Doyle, Low 2001, …

Duality Approach

))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+

Primal-dual algorithm:

−+=

∈∀≤

∑∑

∑

≥≥

≥

)( )( max )( min

, subject to )( max

00

0

:Dual

:Primal

ll

ll

sss

xp

ll

sss

x

xcpxUpD

LlcxxU

s

s

Duality Model of TCP

))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+


Reno, Vegas

DropTail, RED, REM

Source algorithm iterates on ratesLink algorithm iterates on pricesWith different utility functions


))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+


Reno, Vegas

DropTail, RED, REM

(x*,p*) primal-dual optimal if and only if

0 if equality with ** >≤ lll pcy


))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+


Reno, Vegas

DropTail, RED, REM

Any link algorithm that stabilizes queuegenerates Lagrange multiplierssolves dual problem

Gradient algorithm

Gradient algorithm

))(( )1( : source 1' tqUtx iii−=+

+−+=+ )])(()([ )1( :link lllll ctytptp γ

Theorem (Low & Lapsley ’99)

Converge to optimal rates in distributed asynchronous environment

Gradient algorithm

Gradient algorithm

))(( )1( : source 1' tqUtx iii−=+

+−+=+ )])(()([ )1( :link lllll ctytptp γ

Vegas: approximate gradient algorithm

( ))()(sgn ))((

1 )( 2 txtxtqd

txF iiii

ii −+

+=

))(( 1' tqU ii−

Summary: equilibrium

Llcx

xU

l

l

sss

xs

∈∀≤

∑≥

, subject to

)( max0

Flow control problem

TCP/AQM Maximize aggregate source utilityWith different utility functions

Primal-dual algorithm

))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+ Reno, Vegas

DropTail, RED, REM

Implications

PerformanceRate, delay, queue, loss

FairnessUtility function

Persistent congestion

Performance

DelayCongestion measures: end to end queueingdelay

Sets rate

Equilibrium condition: Little’s Law

LossNo loss if converge (with sufficient buffer)Otherwise: revert to Reno (loss unavoidable)

)()(

tqdtxs

sss α=

Vegas Utility

iiiii xdxU log)( α=

Equilibrium (x, p) = (F, G)

Proportional fairness

Validation (L. Wang, Princeton)

Source 1 Source 3 Source 5

RTT (ms) 17.1 (17) 21.9 (22) 41.9 (42) Rate (pkts/s) 1205 (1200) 1228 (1200) 1161 (1200)Window (pkts) 20.5 (20.4) 27 (26.4) 49.8 (50.4)Avg backlog (pkts) 9.8 (10)

NS-2 simulation, single link, capacity = 6 pkts/ms5 sources with different propagation delays, αs = 2 pkts/RTT

meausred theory

Persistent congestionVegas exploits buffer process to compute prices (queueing delays)Persistent congestion due to

Coupling of buffer & priceError in propagation delay estimation

ConsequencesExcessive backlogUnfairness to older sources

Theorem (Low, Peterson, Wang ’02)

A relative error of εs in propagation delay estimation distorts the utility function to

sssssssss xdxdxU εαε ++= log)1()(ˆ


Single link, capacity = 6 pkt/ms, αs = 2 pkts/ms, ds = 10 msWith finite buffer: Vegas reverts to Reno

Without estimation error With estimation error


Source rates (pkts/ms)# src1 src2 src3 src4 src51 5.98 (6)2 2.05 (2) 3.92 (4)3 0.96 (0.94) 1.46 (1.49) 3.54 (3.57)4 0.51 (0.50) 0.72 (0.73) 1.34 (1.35) 3.38 (3.39)5 0.29 (0.29) 0.40 (0.40) 0.68 (0.67) 1.30 (1.30) 3.28 (3.34)

# queue (pkts) baseRTT (ms)1 19.8 (20) 10.18 (10.18)2 59.0 (60) 13.36 (13.51)3 127.3 (127) 20.17 (20.28)4 237.5 (238) 31.50 (31.50)5 416.3 (416) 49.86 (49.80)

Outline





))( ),(( )1())( ),(( )1(txtpGtptxtpFtx

=+=+

Vegas model

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

x y

q p

+

−+= 1)()(

l

lll c

tytpG

( ))()(sgn ))((

1 )( 2 tqtxdtqd

txF iiiiii

ii −+

+= α

Vegas model

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

x y

q p

[ ] lieR lislif link uses source if τv−=

[ ] lieR lislib link uses source if τw−=

Approximate model

( )ii

iid

tqtxi tT

tx α)()(

2 1sgn )(

1 )( −=&

Approximate model

( )ii

iid

tqtxi tT

tx α)()(

2 1sgn )(

1 )( −=&

( )ii

iid

tqtxi tT

tx αηπ

)()(1-2 1 tan

)(12 )( −=&

Linearized model

iii

i

i

ii q

asTa

qxx ∂

+−=∂ & l

ll y

cp ∂−=∂ γ&

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

x y

q p

12 ii

i Txa

πη

−= γ controls equilibrium delay

Stability

Cannot be satisfied with >1 bottleneck link!

Theorem (Choe & Low, ‘02)

Locally asymptotically stable if

c

c

i

i

aa

ωωsin

min

time triprouddelay queueinglink

> > 0.63

Stabilized Vegas

( ))(1 tan)(

12 )( )()(1-2 tq

tTtx iid

tqtxi ii

ii && κηπ α −−=

( )ii

iid

tqtxi tT

tx αηπ

)()(1-2 1 tan

)(12 )( −=&

Linearized model

iii

i

i

ii q

asTa

qxx ∂

+−=∂ & l

ll y

cp ∂−=∂ γ&

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

x y

q p

γ controls equilibrium delay

Linearized model

iii

iii q

asasbx ∂

++

−=∂ µ

& ll

l yc

p ∂−=∂ γ&

F1

FN

G1

GL

Rf(s)

Rb’(s)

TCP Network AQM

x y

q p

γ controls equilibrium delaychoose ai = a, αi = µ

Stability



),( queueing tripround

time tripround µaσM <

exampleLHS < 10*10 = 100a = 0.1, µ = 0.015 σ (a,µ) = 120

Stability



),( queueing tripround

time tripround µaσM <

ApplicationStabilized TCP with current routersQueueing delay as congestion measure has the right scalingIncremental deployment with ECN

Vertical decomposition

Utility maximization

ll

li l

lliRiiixp

ll

iii

xR

cppRxxU

Llcy

xU

ii

i

∑∑ ∑

∑

+

−

∈∀≤

≥≥

≥

min)(max min

, subject to

)( maxmax

00

0

:Dual

:Primal

Shortest path routing!


Utility maximization

ll

li l

lliRiiixp

ll

iii

xR

cppRxxU

Llcy

xU

ii

i

∑∑ ∑

∑

+

−

∈∀≤

≥≥

≥

min)(max min

, subject to

)( maxmax

00

0

:Dual

:Primal

Can shortest-path routing (IP) and TCP/AQM maximize utility?


Theorem (Wang, Li, Low, Doyle ‘02)

Primal problem is NP-hard

Cannot be solved by shortest-path routing and TCP/AQMShortest path routing based on prices can be unstableEven when stable, there can be duality gapHow well does TCP/AQM/IP solve it approximately?

Papers

A duality model of TCP flow controls (ITC, Sept 2000)

Optimization flow control, I: basic algorithm & convergence (ToN, 7(6), Dec 1999)

Understanding Vegas: a duality model (J. ACM, 2002)

Scalable laws for stable network congestion control (CDC, 2001)

REM: active queue management (Network, May/June 2001)

netlab.caltech.edu

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