Optimization Models for Heterogeneous Protocols€¦ · Application 1) Understand each layer in isolation, assuming other layers are designed nearly optimally 2) Understand interactions

Optimization Models for Heterogeneous Protocols

Steven Low

CS, EEnetlab.CALTECH.edu

with J. Doyle, S. Hegde, L. Li, A. Tang, J. Wang, ClatechM. Chiang, Princeton

oInternet protocolsoHorizontal decompositionn TCP-AQMnSome implications

oVertical decompositionn TCP/IP, HTTP/TCP, TCP/wireless, …

oHeterogeneous protocols

Outline

TCP/AQM

IP

Link

Application

cRx

xUi

iix

≤

∑≥

tosubj

)( max0

Internet Protocols

TCP/AQM

IP

xi(t)

pl(t)

Link

Application o Protocols determines network behavioro Critical, yet difficult, to understand and

optimizeo Local algorithms, distributed spatially

and vertically à global behavioro Designed separately, deployed

asynchronously, evolves independently

Internet Protocols

TCP/AQM

IP

xi(t)

pl(t)

Link





Need to reverse engineer

… to understand stack and network as whole

Internet Protocols

TCP/AQM

IP

xi(t)

pl(t)

Link





Need to reverse engineer

… to forward engineer new large networks

Internet Protocols

Minimize path costs (IP)

TCP/AQM

IP

xi(t)

pl(t)

Maximize utility (TCP/AQM)

Link Minimize SIR, max capacities, …

Application Minimize response time (web layout)

o Each layer is abstracted as an optimization problem

oOperation of a layer is a distributed solutiono Results of one problem (layer) are parameters of

othersoOperate at different timescales

Internet Protocols

TCP/AQM

IP

Link

Application

cRx

xUi

iix

≤

∑≥

tosubj

)( max0

o Each layer is abstracted as an optimization problem

oOperation of a layer is a distributed solutiono Results of one problem (layer) are parameters of

othersoOperate at different timescales

TCP/AQM

IP

Link

Application

1) Understand each layer in isolation, assumingother layers are designed nearly optimally2) Understand interactions across layers3) Incorporate additional layers4) Ultimate goal: entire protocol stack as solving one giant optimization problem, where individual layers are solving parts of it

Protocol Decomposition

Layering as Optimization Decomposition

o Layering as optimization decompositionn Network NUM problemn Layers Subproblemsn Layering Decomposition methodsn Interface Primal or dual variables

o Enables a systematic study of: n Network protocols as distributed solutions to global

optimization problemsn Inherent tradeoffs of layeringn Vertical vs. horizontal decomposition

(M. Chiang, Princeton)




Outline

TCP/AQM

IP

Link

Application

cRx

xUi

iix

≤

∑≥

tosubj

)( max0

Network model

F1

FN

G1

GL

R

RT

TCP Network AQM

x y

q p

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

tRxtpGtptxtpRFtx T

=+=+ Reno, Vegas

DT, RED, …

liR li link uses source if 1= IP routing

Network model: example

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

tRxtpGtptxtpRFtx T

=+=+ Reno, Vegas

DT, RED, …


=+

−=+

∑

∑

ililill

llli

i

ii

tptxRGtp

tpRx

Ttx

)(),()1(

)(2

1)1(

2

2

TCP Reno: currently deployed TCP

AI

MD

TailDrop

Network model: example

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

tRxtpGtptxtpRFtx T

=+=+ Reno, Vegas

DT, RED, …


−+=+

−+=+

∑

∑

ilili

lll

llliii

i

iii

ctxRc

tptp

tpRtxT

txtx

)(1

)()1(

)()()()1( αγ

TCP FAST: high speed version of Vegas

n TCP-AQM:

),(

),( ***

***

RxpGp

pRxFx T

=

=

Duality model

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

n F determines utility function UnG determines complementary slackness conditionn p* are Lagrange multipliers

Uniqueness of equilibriumn x* is unique when U is strictly concaven p* is unique when R has full row rank

cRxxU ii

ix

≤∑≥

t.s. )( max0

n TCP-AQM:

Duality model


n F determines utility function UnG determines complementary slackness conditionn p* are Lagrange multipliers

The underlying concave program also leads to simple dynamic behavior

cRxxU ii

ix

≤∑≥

t.s. )( max0

),(

),( ***

***

RxpGp

pRxFx T

=

=

Duality modeln Global stability in absence of feedback delayn Lyapunov functionn Kelly, Maulloo & Tan (1988)

n Gradient projectionn Low & Lapsley (1999)

n Singular perturbationsn Kunniyur & Srikant (2002)

n Passivity approachnWen & Arcat (2004)

n Linear stability in presence of feedback delayn Nyquist criterian Paganini, Doyle, Low (2001), Vinnicombe (2002), Kunniyur

& Srikant (2003)

n Global stability in presence of feedback delayn Lyapunov-Krasovskii, SoSTooln Papachristodoulou (2005)

n Global nonlinear invariance theoryn Ranjan, La & Abed (2004, delay-independent)

Duality model


cRxxU ii

ix

≤∑≥

t.s. )( max0

n α = 1 : Vegas, FAST, STCP n α = 1.2: HSTCP (homogeneous sources)

n α = 2 : Reno (homogeneous sources)

n α = infinity: XCP (single link only)

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU (Mo & Walrand 00)




Outline

TCP/AQM

IP

Link

Application

cRx

xUi

iix

≤

∑≥

tosubj

)( max0

FAST Architecture

Each componento designed independentlyo upgraded asynchronously

Is large queue necessary for high throughput?

Efficient and Friendly

o Fully utilizing bandwidth

o Does not disrupt interactive applications

One-way delay Requirement for VoIP

o DSL upload (max upload capacity 512kbps)o Latency: 10ms

Resilient to Packet Loss

o Lossy link, 10Mbpso Latency: 50ms

Current TCP collapsesat packet loss rate bigger than a few %!

FAST

max

Implications

n Is fair allocation always inefficient ?

n Does raising capacity always increase throughput ?

Intricate and surprising interactions in large-scalenetworks … unlike at single link

Implications

n Is fair allocation always inefficient ?

n Does raising capacity always increase throughput ?

Intricate and surprising interactions in large-scalenetworks … unlike at single link

Daulity model

n α = 1 : Vegas, FAST, STCP n α = 1.2: HSTCP (homogeneous sources)

n α = 2 : Reno (homogeneous sources)

n α = infinity: XCP (single link only)

cRx

xU ii

ix

≤

∑≥

t.s.

)( max0

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU

(Mo, Walrand 00)

Fairness

n α = 0: maximum throughput n α = 1: proportional fairness n α = 2: min delay fairness n α = infinity: maxmin fairness

cRx

xU ii

ix

≤

∑≥

t.s.

)( max0

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU

(Mo, Walrand 00)

Fairness

n Identify allocation with αn An allocation is fairer if its α is larger

cRx

xU ii

ix

≤

∑≥

t.s.

)( max0

(Mo, Walrand 00)

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU

Efficiency: aggregate throughput

n Unique optimal rate x(α) n An allocation is efficient if T(α) is large

cRx

xU ii

ix

≤

∑≥

t.s.

)( max0

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU

∑=i

ixT )(:)( throughput αα

Conjecture

ConjectureT(α) is nonincreasing

i.e. a fair allocation is always inefficient

Example 1



1=lc0

max throughput

1

1/(L+1)

proportionalfairness

L/L(L+1)

)()1( )0( ∞>> TTT

1/2

maxminfairness

1/2

Example 1



1/1

/1

+α

α

LL

11

/1 +αL1=lc

Example 2

21 cc ≤

−+++= 21

22

21213

2)1( ccccccT

2)( 2

1

ccT +=∞

)()1( ∞>⇒ TT



Example 3

↓)(αT



Intuition

“The fundamental conflict between achieving flow fairness and maximizing overall system throughput….. The basic issue is thus the trade-off between these two conflicting criteria.”

Luo,etc.(2003), ACM MONET

Results

oTheorem: Necessary & sufficient condition for general networks (R, c) provided every link has a 1-link flow

oCorollary 1: true if N(R)=1

1/1

/1

+α

α

LL

11

/1 +αL1=lc

Results


oCorollary 1: true if N(R)=1

21 cc ≤)()1( ∞>⇒ TT

Results


oCorollary 2: true ifnN(R)=2n2 long flows pass through same# links

Counter-example

o Theorem: Given any α0>0, there exists network where

oCompact example

0α

0 allfor 0 ααα

>>ddT

Counter-example

oThere exists a network such thatdT/dα > 0 for almost all α>0

oIntuitionnLarge α favors expensive flowsnLong flows may not be expensive

oMax-min may be more efficient than proportional fairness

expensive long

Efficiency: aggregate throughput

n Unique optimal rate x(α; c) n An allocation is efficient if T(α ; c) is large

cRx

xU ii

ix

≤

∑≥

t.s.

)( max0

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU

∑=i

i cxcT );(:);( throughput αα

Throughput & capacity

oIntuition: Increasing link capacities always raises throughput T

oTheorem: Necessary & sufficient condition for general networks (R, c)

oCorollary: For all α, increasingn a link’s capacity can reduce Tn all links’ capacities equally can reduce Tn all links’ capacities proportionally raises T


n

n

cRx

xU ii

ix

≤

∑≥

t.s.

)( max0

=≠−

=−

1 if log1 if )1(

)(1

ααα α

i

iii x

xxU

∑=i

i cxcT );(:);( Throughput αα

δε

εδααδα

ε cxcTcT

DT T

∂∂

=+−

=→

1 ),();(

lim:);(0


n

n

∑=i

i cxcT );(:);( Throughput αα

δε

εδααδα

ε cxcTcT

DT T

∂∂

=+−

=→

1 ),();(

lim:);(0

Corollary:

oGiven any α0>0, there exists network (R, c) s.t. for all α>α0

nDT(α; e1) < 0 for some lnDT(α; 1) < 0

oFor all networks (R, c) and for all α>0nDT(α; c) > 0




Outline

TCP/AQM

IP

Link

Application

cRx

xUi

iix

≤

∑≥

tosubj

)( max0

Protocol decomposition

TCP-AQM:n TCP algorithms maximize utility with different

utility functions

Congestion prices coordinate across protocol layers

BccRx

xU

T

iii

xRc

≤≤

∑≥

α , subject to

)( maxmax max0

TCP-AQM

IP

TCP/AQM

Applications

Link


TCP-AQMIP

BccRx

xU

T

iii

xRc

≤≤

∑≥

α , subject to

)( maxmax max0

TCP-AQM

IP

TCP/AQM

Applications

Link

TCP/IP:n TCP algorithms maximize utility with different

utility functionsn Shortest-path routing is optimal using congestion

prices as link costs ……


Two timescales

n Instant convergence of TCP/IPn Link cost = a pl(t) + b dl

n Shortest path routing R(t)price

static

TCP/AQM

IPR(0)

a p(0)

R(1)

a p(1)

… R(t), R(t+1) ,…

TCP-AQM/IP Model

TCP

cxtR

xUtxi

iix

≤

= ∑≥

)( subject to

)( max arg )(0

∑

∑ ∑

+

−=

≥≥

lll

i llliiiixp

pc

ptRxxUtpi

)()(max min arg )(00

AQM

))((minarg)1( lll

liRi bdtapRtRli

+=+ ∑IP

Link cost

Questions

nDoes equilibrium routing Ra exist ?nWhat is utility at Ra?nIs Ra stable ? nCan it be stabilized?

TCP/AQM

IPR(0)

a p(0)

R(1)

a p(1)

… R(t), R(t+1) ,…

Equilibrium routing

Theorem1. If b=0, Ra exists iff zero duality gapn Shortest-path routing is optimal with

congestion pricesn No penalty for not splitting

+

−

≤

∑∑ ∑

∑

≥≥

≥

ll

li l

lliRiiixp

iii

xR

cppRxxU

cRxxU

ii

max)( max min

subject to )( maxmax

00

0

:Dual

:Primal


TCP-AQMIP

TCP/IP (with fixed c):n Equilibrium of TCP/IP exists iff zero duality gapn NP-hard, but subclass with zero duality gap is P n Equilibrium, if exists, can be unstablen Can stabilize, but with reduced utility

Inevitable tradeoff bw utility max & routing stability

BccRx

xU

T

iii

xRc

≤≤

∑≥

α , subject to

)( maxmax max0

TCP-AQM

IP

TCP/AQM

Applications

Link


IPLink

TCP/IP with optimal c: n With optimal provisioning, static routing is optimal using

provisioning cost α as link costs

TCP/IP with static routing in well-designed network

TCP-AQMIP

BccRx

xU

T

iii

xRc

≤≤

∑≥

α , subject to

)( maxmax max0

TCP-AQM

IP

TCP/AQM

Applications

Link

Summary

BccRx

xU

T

iii

xRc

≤≤

∑≥

α , subject to

)( maxmax max0

TCP-AQMIPLink

n TCP algorithms maximize utility with different utility functions

n IP shortest path routing is optimal using congestion prices as link costs, with given link capacities c

n With optimal provisioning, static routing is optimal using provisioning cost α as link costs





Outline

TCP/AQM

IP

Link

Application

cRx

xUi

iix

≤

∑≥

tosubj

)( max0

Congestion control

F1

FN

G1

GL

R

RT

TCP Network AQM

x y

q p

=+

=+

∑

∑

iililll

il

lliii

txRtpGtp

txtpRFtx

)( ),( )1(

)( ,)( )1(same price

for all sources

Heterogeneous protocols

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


13M61Mpath 293M27Mpath 3

13M52MPath 1

eq 2eq 1eq 1

eq 2

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


13M61Mpath 293M27Mpath 3

13M52MPath 1

eq 2eq 1eq 1

eq 2

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

eq 3 (unstable)

>=≤

=

∑

∑

0 if

)(

)(

, ll

lji

ji

jli

ll

jl

jli

ji

ji

pcc

pxR

pmRF(p)x

Multi-protocol: J>1

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

multiplier

n TCP-AQM equilibrium p:

n TCP-AQM equilibrium p:

>=≤

=

∑

∑

0 if

)(

)(

, ll

lji

ji

jli

ll

jl

jli

ji

ji

pcc

pxR

pmRF(p)x

Multi-protocol: J>1

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

Results: existence of equilibrium

n Equilibrium p always exists despite lack of underlying utility maximization

n Generally non-uniquen Network with unique bottleneck set but

uncountably many equilibrian Network with non-unique bottleneck sets

each having unique equilibrium

Results: regular networks

nRegular networks: all equilibria p are locally unique

Theorem (Tang, Wang, Low, Chiang, Infocom 2005)

nAlmost all networks are regularnRegular networks have finitely many

and odd number of equilibria (e.g. 1)

Proof: Sard’s Theorem and Index Theorem

Results: global uniqueness


nIf all equilibria p have I(p) = (-1)L

then p is globally unique nIf all equilibria p all locally stable, then

it is globally unique

( )( )

><−

=

∂∂== ∑

0det if 1 0det if 1

: )(index

)(: )(:)(,

(p)(p)

pI

ppy

(p)pxRpy ji

ji

jli

JJ

J



nFor J=1, equilibrium p is globally unique if R is full rank (Mo & Walrand ToN 2000)

nFor J>1, equilibrium p is globally unique if J(p) is negative definite over a certain set

)(: )(:)(,

ppy

(p)pxRpy ji

ji

jli ∂

∂== ∑ J



nIf price mapping functions mlj are `similar’,

then equilibrium p is globally unique

nIf price mapping functions mlj are linear and

link-independent, then equilibrium p is globally unique

0any for ]2,[

0any for ]2,[/1

/1

>∈

>∈jjLjj

l

llL

lj

l

aaam

aaam

&

&

Summary: equilibrium structure

Uni-protocolnUnique bottleneck

setn Unique rates &

prices

Multi-protocoln Non-unique bottleneck

sets n Non-unique rates &

prices for each B.S.

n always oddn not all stablen uniqueness

conditions

Experiments: non-uniqueness

Discovered examples guided by theoryn Numerical examples n NS2 simulations (Reno + Vegas)n DummyNet experiments (Reno + FAST)n Practical network??

Documents

Optimization Models for Heterogeneous Protocols€¦ · Application 1) Understand each layer in isolation, assuming other layers are designed nearly optimally 2) Understand interactions