Heterogeneous Congestion Control Protocols

Steven Low

CS, EEnetlab.CALTECH.edu

with A. Tang, J. Wang, D. Wei, CaltechM. Chiang, Princeton

Outline

Review: homogeneous case Motivating experiments Model Equilibrium

Existence, uniqueness, local stability Slow timescale control

Tang, Wang, Low, Chiang. Infocom March 2005Tang, Wang, Hegde, Low. Telecommunications Systems, Dec 2005

F1

FN

G1

GL

R

RT

TCP Network AQM

x y

q p

))( ),(( )1(

))( ),(( )1(

tRxtpGtp

txtpRFtx T

Reno, Vegas

DT, RED, …

liRli link uses source if 1 IP routing

Network model

for every RTT{ W += 1 }for every loss{ W := W/2 }

Reno: (AI)

(MD)Jacobson

1989

ililill

llli

i

ii

tptxRGtp

tpRx

Ttx

)(),()1(

)(2

1)1(

2

2

AI

MD

TailDrop

Network model: example

FAST:Jin, Wei, Low

2004 W

RTT

baseRTT :W

periodically{

}

ilili

lll

llliii

i

iii

ctxRc

tptp

tpRtxT

txtx

)(1

)()1(

)()()()1(

Network model: example

Duality model of TCP/AQM TCP/AQM

Equilibrium (x*,p*) primal-dual optimal:

F determines utility function U G guarantees complementary slackness p* are Lagrange multipliers

) ,(

) ,( ***

***

RxpGp

xpRFx T

cRxxU iix

subject to )( max0

Uniqueness of equilibrium x* is unique when U is strictly

concave p* is unique when R has full row rank

Kelly, Maloo, Tan 1998Low, Lapsley 1999

Duality model of TCP/AQM TCP/AQM

Equilibrium (x*,p*) primal-dual optimal:

F determines utility function U G guarantees complementary slackness p* are Lagrange multipliers

) ,(

) ,( ***

***

RxpGp

xpRFx T

cRxxU iix

subject to )( max0

Kelly, Maloo, Tan 1998Low, Lapsley 1999

The underlying concave program also leads to simple dynamic behavior

Duality model of TCP/AQM

Equilibrium (x*,p*) primal-dual optimal:

cRxxU iix

subject to )( max0

Mo & Walrand 2000:

1 if )1(

1 if log)(

11

i

i

iix

xxU

Vegas, FAST, STCP

HSTCP Reno XCP (single link

only)

Low 2003

Some implications Equilibrium

Always exists, unique if R is full rank Bandwidth allocation independent of AQM or arrival Can predict macroscopic behavior of large scale

networks Counter-intuitive throughput behavior

Fair allocation is not always inefficient Increasing link capacities do not always raise

aggregate throughput [Tang, Wang, Low, ToN

2006]

FAST TCP Design, analysis, experiments

[Jin, Wei, Low, ToN 2007]

Some implications Equilibrium

Always exists, unique if R is full rank Bandwidth allocation independent of AQM or arrival Can predict macroscopic behavior of large scale

networks Counter-intuitive throughput behavior

Fair allocation is not always inefficient Increasing link capacities do not always raise

aggregate throughput [Tang, Wang, Low, ToN

2006]

FAST TCP Design, analysis, experiments

[Jin, Wei, Low, ToN 2007]

Outline

Review: homogeneous case Motivating experiments Model Equilibrium

Existence, uniqueness, local stability Slow timescale control

Throughputs depend on AQM

FAST and Reno share a single bottleneck router NS2 simulation Router: DropTail with variable buffer size With 10% heavy-tailed noise traffic

FAST throughput

buffer size = 80 pkts buffer size = 400 pkts

Multiple equilibria: throughput depends on arrival

eq 1 eq 2

Path 1 52M 13M

path 2 61M 13M

path 3 27M 93M

eq 1

eq 2

Tang, Wang, Hegde, Low, Telecom Systems, 2005

Dummynet experiment

eq 1 eq 2

Path 1 52M 13M

path 2 61M 13M

path 3 27M 93MTang, Wang, Hegde, Low, Telecom Systems, 2005

eq 1

eq 2 eq 3 (unstable)

Dummynet experiment

Multiple equilibria: throughput depends on arrival

Some implications

homogeneous heterogeneous

equilibrium unique non-unique

bandwidthallocation on AQM

independent dependent

bandwidthallocationon arrival

independent dependent

Duality model:

, ***

lilliii xpRFxcRxxU ii

x

s.t. )( max

0

llliii

i

iii pRx

TxF

Why can’t use Fi’s of FAST and Reno in duality model?

l

llii

ii pR

x

TF

2

1 2

2

delay for FAST

loss for Reno

They use different prices!

Duality model:

, ***

lilliii xpRFxcRxxU ii

x

s.t. )( max

0

llliii

i

iii pRx

TxF

Why can’t use Fi’s of FAST and Reno in duality model?

l

llii

ii pR

x

TF

2

1 2

2

They use different prices!

ilili

ll ctxR

cp )(

1

iililll txRtpgp )(),(

F1

FN

G1

GL

R

RT

TCP Network AQM

x y

q p

iililll

il

lliii

txRtpGtp

txtpRFtx

)( ),( )1(

)( ,)( )1(

same pricefor all sources

Homogeneous protocol

F1

FN

G1

GL

R

RT

TCP Network AQM

x y

q p

)( ,)( )1(

)( ,)( )1(

txtpmRFtx

txtpRFtx

ji

ll

jlli

ji

ji

il

lliii heterogeneousprices for

type j sources

Heterogeneous protocol

Heterogeneous protocols

Equilibrium: p that satisfies

i,j ll

lji

jlil

ll

jlli

ji

ji

pc

cpxRpy

pmRfpx

0 if

)( : )(

)( )(

Duality model no longer applies ! pl can no longer serve as Lagrange

multiplier

Heterogeneous protocols

Equilibrium: p that satisfies

i,j ll

lji

jlil

ll

jlli

ji

ji

pc

cpxRpy

pmRfpx

0 if

)( : )(

)( )(

Need to re-examine all issues Equilibrium: exists? unique? efficient? fair? Dynamics: stable? limit cycle? chaotic? Practical networks: typical behavior? design

guidelines?

Heterogeneous protocols

Equilibrium: p that satisfies

i,j ll

lji

jlil

ll

jlli

ji

ji

pc

cpxRpy

pmRfpx

0 if

)( : )(

)( )(

Dynamic: dual algorithm

llll

ll

jlli

ji

ji

ctpyp

tpmRftpx

))((

))(( ))((

Notation

Simpler notation: p is equilibrium if

on bottleneck links

Jacobian:

Linearized dual algorithm:

cpy )(

)( : )( pp

yp

J

p(t)pp )( *J

See Simsek, Ozdaglar, Acemoglu 2005for generalization

Outline

Review: homogeneous case Motivating experiments Model Equilibrium

Existence, uniqueness, local stability Slow timescale control

Tang, Wang, Low, Chiang. Infocom 2005

Existence

Theorem

Equilibrium p exists, despite lack of underlying utility maximization

Generally non-unique There are networks with unique bottleneck

set but infinitely many equilibria There are networks with multiple bottleneck

set each with a unique (but distinct) equilibrium

Regular networks

Definition

A regular network is a tuple (R, c, m, U) for which all equilibria p are locally unique, i.e.,

Theorem Almost all networks are regular A regular network has finitely many and

odd number of equilibria (e.g. 1)

Proof: Sard’s theorem and index theorem

0 )(det : )(det

pp

ypJ

Global uniqueness

Corollary If price mapping functions ml

j are linear and link-independent, then equilibrium is globally unique

e.g. a network of RED routers almost always has globally unique equilibrium

Theorem If price heterogeneity is small, then equilibrium is

globally unique

0any for ]2,[

0any for ]2,[/1

/1

jjLjj

l

llL

lj

l

aaam

aaam

Local stability:`uniqueness’ stability

Theorem If price heterogeneity is small, then the unique

equilibrium p is locally stable

0any for ]2,[

0any for ]2,[/1

/1

jjLjj

l

llL

lj

l

aaam

aaam

p(t)pp )( *JLinearized dual algorithm:

Equilibrium p is locally stable if

0 )( Re pJ

Local stability:`converse’

Theorem If all equilibria p are locally stable, then it is globally

unique

LpI )1( )( Proof idea: For all equilibrium p: Index theorem:

L

p

pI )1( )( eq