ELE539A: Optimization of Communication Systems

Lecture 11: Layering As Optimization Decomposition

I

Professor M. Chiang

Electrical Engineering Department, Princeton University

March 1, 2006

Overview

• Background: NUM and G.NUM

• Summary: Key Messages

• Part I: Horizontal Decompositions

• Part II: Vertical Decompositions

• Future Research: Stochastic and Nonconvex

Acknowledgement: Rob Calderbank, Lijun Chen, John Doyle, Jiayue

He, Jang-Won Lee, Steven Low, Daniel Palomar, Jennifer Rexford

Nature of the Talk

• What’s covered:

Give an overview of the topic. Details in various papers

• What’s not covered:

Not exhaustive survey. Highlight the key ideas and challenges

• What’s emphasized:

Biased presentation. Focus on work by my collaborators and me

Layered Network Architecture

Application

Presentation

Session

Transport

Network

Link

Physical

Important foundation for data networking

Ad hoc design historically (within and across layers)

Layering As Optimization Decomposition

How to, and how not to, layer? A question on structure

The first unifying view and systematic approach

Network: Generalized NUM

Layering architecture: Decomposition scheme

Layers: Decomposed subproblems

Interfaces: Functions of primal or dual variables

Rigorous math (optimization and distributed algorithm)

Versatile applications (horizontal and vertical decompositions) through

• implicit message passing

• explicit message passing

Layering as Optimization Decomposition

What’s so unique about this framework for cross-layer design?

• Network as optimizer

• End-user application utilities as the driver

• Performance benchmark without any layering

• Unified approach to cross-layer design

• Systematic exploration of architectural alternatives

Not every cross-layer paper is ‘layering as optimization decomposition’

Practical Side

Industry adoption of ‘layering as optimization decomposition’:

• Internet resource allocation: TCP FAST (Caltech)

• Protocol stack design: Internet 0 (MIT)

• Broadband access: FAST Copper (Princeton, Stanford, Fraser)

This talk is about the underlying common language, methodologies,

and specific designs

Connections with Mathematics

• Convex and nonconvex optimization

• Decomposition and distributed algorithm

• Algebraic geometry

• Differential topology

• Game theory, General market equilibrium theory

References: Overview

Overview: M. Chiang, S. H. Low, A. R. Calderbank, and J. C. Doyle,

“Layering as optimization decomposition,” Proceedings of IEEE, 2006.

Introduction: KMT98: F. P. Kelly, A. Maulloo, and D. Tan, “Rate

control for communication networks: shadow prices, proportional

fairness and stability,” Journal of Operations Research Society, vol. 49,

no. 3, pp.237-252, March 1998.

Introduction: Chi05 M. Chiang, “Balancing transport and physical layer

in wireless multihop networks: Jointly optimal congestion control and

power control,” IEEE J. Sel. Area Comm., vol. 23, no. 1, pp. 104-116,

Jan. 2005.

References: Reverse Engineering

Part IA: Low03 S. H. Low, “A duality model of TCP and queue

management algorithms,” IEEE/ACM Tran. on Networking, vol. 11,

no. 4, pp. 525-536, Aug. 2003.

Part IA: TWLC05 A. Tang, J. Wang, S. H. Low, and M. Chiang,

“Equilibrium of heterogeneous congestion control protocols”, Proc.

IEEE INFOCOM, March 2005.

Part IB: GSW02: T. G. Griffin, F. B. Shepherd, and G. Wilfong, “The

stable path problem and interdomain routing,” IEEE/ACM Trans. on

Networking, vol. 10, no. 2, pp. 232-243, April 2002.

Part IC: LCC06a J. W. Lee, M. Chiang, and R. A. Calderbank,

“Utility-optimal medium access control: Reverse and forward

engineering,” IEEE INFOCOM, 2006.

References: Forward Engineering

Part IIA: WLLD05 J. Wang, L. Li, S. H. Low, and J. C. Doyle, “Can

TCP and shortest path routing maximize utility,” IEEE/ACM Trans. on

Networking, vol. 13, no. 4, Aug 2005.

Part IIA: HCR06 J. He, M. Chiang, J. Rexford, “Stability and optimality

of TCP/IP interactions,” Proc. IEEE ICC, June 2006.

Part IIB: LCC06b J. W. Lee, M. Chiang, and R. A. Calderbank,

“Distributed algorithms for optimal rate-reliability tradeoff in networks,”

To appear IEEE J. Sel. Area Comm., 2006.

Part IIC: WK05 X. Wang and K. Kar, “Cross-layer rate control for

end-to-end proportional fairness in wireless networks with random

access,” Proc. ACM Mobihoc, 2005.

Part IIC: LCC06c J. W. Lee, M. Chiang, and R. A. Calderbank, “Jointly

optimal congestion and medium access control in ad hoc networks”, To

appear IEEE Comm. Lett., 2006.

Part IID: CLCD06: L. Chen, S. H. Low, M. Chiang, and J. C. Doyle, “

Joint optimal congestion control, routing, and scheduling in wirelss ad

hoc networks,” Proc. IEEE INFOCOM, 2006.

Part IIE: PC06: D. Palomar and M. Chiang, “Alternative

decompositions for network utility maximization: Framework and

applications,” Proc. IEEE INFOCOM, 2006.

References: Related Work

A partial list:

Cruz, Santhanam 2003 Infocom

Xiao, Johansson, Boyd 2004 TComm

Chang, Liu 2004 ToN

Neely, Modiano, Rohrs 2005 JSAC

Lin, Shroff 2005 Infocom

Chen, Low, Doyle 2005 Infocom

Erilmaz, Srikant 2005 Infocom

Yu, Yuan 2005 ICC

Xi, Yeh 2005 CISS

Zheng and Zhang 2006 CISS

Layers

Restriction: we focus on resource allocation functions

• TCP: congestion control

Different meanings:

• Routing: RIP/OSPF, BGP, wireless routing, optical routing,

dynamic/static, single-path/multi-path, multicommodity flow routing...

• MAC: scheduling or contention-based

• PHY: power control, coding, modulation, antenna signal processing...

Insights on both:

• What each layer can see

• What each layer can do

Utility

Why utility? (function of rate, useful information, delay, energy...)

• Reverse engineering: TCP maximizes utilities

• Behavioral model: user satisfaction

• Traffic model: traffic elasticity

• Economics: resource allocation efficiency

• Economics: different utility functions define different fairness

• Goal: Distributed algorithm converging to globally and jointly

optimum resource allocation

• Limitations to be discussed at the end

Network Utility Maximization

Basic NUM (KMT98):

maximizeP

s Us(xs)

subject to Rx � c

x � 0

Generalized NUM (one possibility shown here) (Chi05):

maximizeP

s Us(xs, Pe,s) +P

j Vj(wj)

subject to Rx � c(w,Pe)

x ∈ C1(Pe)

x ∈ C2(F)

R ∈ R

F ∈ F

w ∈ W

Dual-based Distributed Algorithm

Basic NUM with concave smooth utility functions:

Convex optimization (Monotropic Programming) with zero duality gap

Lagrangian decomposition:

L(x, λ) =X

s

Us(xs) +X

l

λl

0

@cl −X

s:l∈L(s)

xs

1

A

=X

s

2

4Us(xs) −

0

@

X

l∈L(s)

λl

1

Axs

3

5+X

l

clλl

=X

s

Ls(xs, λs) +X

l

clλl

Dual problem:

minimize g(λ) = L(x∗(λ), λ)

subject to λ � 0

Dual-based Distributed Algorithm

Source algorithm:

x∗s(λs) = argmax [Us(xs) − λsxs] , ∀s

• Selfish net utility maximization locally at source s

Link algorithm (gradient or subgradient based):

λl(t + 1) =

2

4λl(t) − α(t)

0

@cl −X

s:l∈L(s)

xs(λs(t))

1

A

3

5

+

, ∀l

• Balancing supply and demand through pricing

Certain choices of step sizes α(t) of distributed algorithm guarantee

convergence to globally optimal (x∗, λ∗)

Key Messages

Reverse Engineering Forward Engineering

Horizontal Decomposition Part I Not in this talk

Vertical Decomposition Not much known Part II

1. Layers 1-4 can be reverse-engineered as cooperative/non-cooperative

optimization problems. Justification of optimization/game models

2. Many possibilities for alternative decompositions

Key Messages

3. ‘Layering as optimization decomposition’ illustrates the opportunities

and risks of cross-layer interactions in a unifying framework

4. Dual decomposition with distributed subgradient is the most

common way for vertical decomposition, but not the only alternative

5. Congestion price may be the best layering price, but not always

6. Coupling (in objective and constraints) : decoupling in various ways

to enable distributed solution

7. Non-convexity (in objective and constraints): find conditions to

guarantee convexity to enable convergence to global optimum

8. Stochastic issues are important and still under-explored

I: Horizontal Decompositions

Focus on reverse engineering in this talk:

• Layer 4 TCP congestion control: Basic NUM

• Layer 3 IP inter-AS routing: Stable Paths Problem

• Layer 2 MAC backoff contention resolution: Non-cooperative Game

Forward engineering for horizontal decompositions also carried out

recently

Reverse Engineering: TCP

Different source algorithms update primal variables (source rates) for

different utilities (Low03):

• TCP Vegas: log utilities

• TCP Tahoe: arctan utilities

Different queue management update dual variables (link prices):

• FIFO

• RED

Rigorous and significant implications to equilibrium properties of

efficiency, fairness, stability, delay of rate allocation

Topology and Heterogeneous TCP

Can the number of equilibrium be:

0? No

1? Maybe, can check with suffi-

cient conditions

2? Almost never

3? Maybe

∞? Almost never

Poincare-Hopf Index Theorem is

the key!0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

p1

p 2

Reverse Engineering: IP

Well-known problem and solution:

• Intra-AS routing: IGP

• Shortest path problem

Recent reverse-engineering:

• Inter-AS routing: BGP

• Stable paths problem (GSW02)

Reverse Engineering: Random-Access MAC

Binary exponential backoff (BEB) protocol in IEEE 802.11 standard

Non-cooperative game-theoretic model (LCC06a)

• implicitly maximizes a selfish local utility at each link (in the form of

expected net reward for successful transmission)

• uses a stochastic subgradient

Nash equilibrium property:

• existence: guaranteed

• uniqueness: sufficient conditions (on user density and backoff

aggressiveness)

• convergence by best response strategy: sufficient conditions

• social welfare optimality: not attained (due to inadequate feedback

mechanism in the BEB protocol)

II: Vertical Decompositions

• Four case studies:

TCP/IP-IGP (WLLD05,HCR06)

TCP/PHY-coding (LCC06b)

TCP/Random-Access-MAC (WK05,LCC06c)

TCP/Routing/Scheduling-MAC (CLCD06)

• Several other case studies not covered

• Alternative decomposition possibilities outlined (PC06)

TCP/IP

Math: Primal problem of a generalized NUM over R and x:

maxR∈R

maxx�0

X

i

Ui(xi) s. t. Rx � c

Dual problem

minλ�0

X

i

maxxi≥0

Ui(xi) − xi minri

X

l

Rliλl

!

+X

l

clλl

ri is the ith column of R with ril

= Rli

Engineering: TE and user adaptation are coupled

Is TCP/IP (with purely dynamic routing) solving the above problems?

Yes, if TCP/IP equilibrium exists (WLLD05)

TCP/IP

Three time-scales possible (HCR06)

WLLD05 covers part of one time-scale case

Stability analysis (for ring topology) followed by optimality simulation� �� � � �

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) * + , .

TCP/IP: Stability

System Model 1: / 0 1 23 24 0

5 2 06 7 87 9 : ; < = > < ?@ AB C DE A FG F

HIJKILMNOPIQ

R FB C FE A F S A T U V C A S AE A FW 87 9 : ; < = > < ?

@ AB C DE A FG FStable iff initial routing is minimum-hop routing

TCP/IP: Stability

System Model 2:X Y Z [

\ [] Y

^ [ Y_̀ ab̀ c d e f g e hi jk l mn j op o

qrstruvwxyrz { ok l on j o | j } ~ � � � � � �� � � � � � � � �� � �� �� � �� � � �

} � � � �� � � � � � � � � � � � �� � �� �� � �� � � � � � �� � � � � � � � � � � � � � � � ��

Stable if step size is sufficiently small

TCP/IP: Stability

System Model 3: � � � �� �� �   � �

¡ ¢£ ¤ ¥¦ ¢ §¨ §©ª«¬ª­®¯°±

ª²³ § £ ¤ §¦ ¢ §́ ¢

¢́µ ¶ ·̧ ¹ £ º́ ¢µ ¶ ¹ » ¼ ½ ¾ ¢ » ¡ ¢ ¿ À ·Á  à ¤ ¢́ ¢¦ ¢ §Stable if step size is sufficiently small and one link capacity is

sufficiently large

TCP/IP: Optimality

Ring topology simulation:

10−2

10−1

100

101

102

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

capacity of link one

aggr

egat

e ut

ility

gap

System Model OneSystem Model TwoSystem Model Three

10−2

10−1

100

101

102

−2

−1.5

−1

−0.5

0

0.5

capacity of link one

aggr

egat

e ut

ility

gap

System Model OneSystem Model TwoSystem Model Three

TCP/IP: Optimality

Access-core topology simulation:

ÄÅ

ÆÇ

ÈÉ

0 10 20 30 40 50−26

−24

−22

−20

−18

−16

−14

−12

−10

−8

−6

standard deviation of link capacity

aggr

egat

e ut

ility

gap

TCP/IP

Some observations:

• Timescale matters

• Homogeneity helps with optimality

• Interaction based on congestion price may not be the best

Possible solutions:

• Use a static component in link metric

• Centrally minimize a network-wide cost function based on link

utilization

TCP/PHY (Channel Coding)

Intuition:

• Source tradeoff: Higher rate, lower quality

• Link tradeoff: Fatter pipe, lower reliability

Signal quality and physical layer entirely missing from basic NUM

Rate-reliability tradeoff. NUM in (xs, Rs, rl,s):

maximizeP

s Us(xs, Rs)

subject to Rs = 1 −P

l∈L(s) El(rl,s), ∀sP

s∈S(l)xs

rl,s≤ Cmax

l, ∀l

How to distributively find the network-wide, globally optimal tradeoff?

TCP/PHY (Channel Coding)

Joint design algorithm (LCC06b):

• Source problem: maximize total net utility on

rate (with total congestion price) and

reliability (with signal quality price)

• Source algorithm:

local solution of source problem (2 variables)

updates signal quality price

• Network problem: maximize net revenue:

receive revenue from rate

pay price for unreliability

• Link algorithm: update link congestion price

TCP/PHY (Channel Coding)

Differentiated or integrated policies

Source s Link l

sx

sµ

lλ∑∈

=)(sLl

ls λλ

lr∑∈

−=)(

)(1sLl

lls rER

∑∈

=)(lSs

sl µµ

∑∈

=)(lSs

sl xx

Network

Network

sR

• First, Problem transformation

• Second, Dual decomposition based distributed algorithm

TCP/PHY (Channel Coding)

• Introducing auxiliary variable and then log change of variable

decouples the coupling

• But turns objective function into nonconcave function

Algorithm converges to globally optimal rate-reliability tradeoff if

• channel codes are strong enough

• and utilities curved enough

d2Us(xs)

dx2s

≤ −dUs(xs)

xsdxs

TCP/PHY (Channel Coding)

0.1 0.15 0.2 0.25 0.3 0.350.9

0.92

0.94

0.96

0.98

1

Data rate (Mbps)

Rel

iabi

lity

User 2 (diff.)User 2 (int.)User 2 (stat.)

v=0

v=0.5

0 0.1 0.2 0.3 0.4 0.52.5

3

3.5

4

4.5

5

5.5

6

v

Net

wor

k ut

ility

Diff. reliability

Int. reliability

Stat. reliability

TCP/MAC

3 4A B C D E

G

1 2

F

I H

5

6

d d d d

d

d

maximizeP

s Us(xs)

subject toP

s∈S(l) xs ≤ clpl

Q

k∈NIto(l)(1 − P k), ∀l

P

l∈Lout(n) pl = P n, ∀n

0 ≤ pl ≤ 1, ∀l,

Readily decomposable for log utility (WK05)

Generalize to any concave utility function for sufficiently concave utility

functions (LCC06c)

TCP/MAC

• Similar techniques as in TCP/PHY-Coding

• But a primal penalty function approach rather than dual relaxation

approach

TCP and MAC operate on the same timescale here

Could also operate on different timescales under an alternative

decomposition

• Spatially separated

• Functionally glued

• Temporally concurrent

TCP/MAC

Link persistence probability update:

pl(t + 1) =

»

p(t) + α(t)∂V (p,x′)

∂pl

|p=p(t),x′=x′(t)

–1

0

, ∀l

where

∂V (p,x′)

∂pl

= κ

0

@

ǫl

pl

−

P

k∈LIfrom

(tl)ǫk

1 −P

m∈Lout(tl)pm

− δtl

1

A

and

ǫl =

8

>

<

>

:

0, ifX

n∈S(l)

ex′

n ≤ clpl

Y

k∈NIto(l)

(1 −X

m∈Lout(k)

pm)

1, otherwise

δn =

8

<

:

0, ifP

m∈Lout(n) pm ≤ 1

1, otherwise.

TCP/MAC

Source rate update:

x′s(t + 1) =

»

x′s(t) + α(t)

∂V (p,x′)

∂x′s

|p=p(t),x′=x′(t)

–x′maxs

x′mins

, ∀s

where

∂V (p,x′)

∂x′s

=∂U ′

s(x′s)

∂x′s

− κX

l∈L(s)

ǫl

ex′

s

P

k∈S(l) ex′

k

TCP/MAC

3 4A B C D E

G

1 2

F

I H

5

6

d d d d

d

d

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

0.4

0.5

Time

Per

sist

ence

pro

babi

lity

Link 1Link 2Link 3Link 4Link 5Link 6

TCP/MAC

2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

α

Dat

a ra

te

Link 1

Link 2

Link 3

Link 4

Link 5

Link 6

2 4 6 8 100

1

2

3

4

5

6

7

8

α

Dat

a ra

te

Source 1Source 2Source 3Total

α-fair utility function: U(x) = x1−α

1−α

TCP/Routing/Scheduling

maximizeP

s Us(xs)

subject to xki ≤

P

j:(i,j)∈L fkij −

P

j:(j,i)∈L fkji, ∀i, j, k

f ∈ Π

Optimal joint design algorithm (CLCD06):

1. Congestion price and source rate update as before

2. For each link (i, j), find destination k∗ such that

k∗ ∈ arg maxk λki − λk

j , and define w∗ij := λk∗

i − λk∗

j . Scheduling:

choose an f̃ ∈ arg maxf∈ΠP

(i,j)∈L w∗ijfij .

3. Routing: over link (i, j) ∈ L, send an amount of data for destination

k∗ according to the rate determined by the above schedule.

Extended to time-varying channels and remain stable

Alternative Decompositions

Many ways to decompose:

• Primal and dual decomposition

• Partial decomposition

• Multi-level decomposition

Master Problem

Subproblem 1...

Secondary Master Problem

prices / resources

First LevelDecomposition

Second LevelDecomposition

Subproblem

Subproblem N

prices / resources

Different communication overhead, computation distribution,

convergence behavior

Primal Decomposition

Simple example:

x + y + z + w ≤ c

Decomposed into:

x + y ≤ α

z + w ≤ c − α

New variable α updated by various methods

Interpretation: Direct resource allocation (rather than pricing-based

control)

Alternative Decompositions

Systematically explore the space of alternative decompositions (PC06)

minimizeP

i Ui(xi)

subject to fi(xi, yi) = 0, ∀i,P

i gi(xi, yi) ≤ 1

5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

Evolution of λ4 for all methods

iteration

Method 1 (subgradient)Method 2 (Gauss−Seidel for all lambdas and gamma)Method 3 (Gauss−Seidel for each lambda and gamma sequentially)Method 4 (subgradient for gamma and exact for all inner lambdas)Method 5 (subgradient for all lambdas and exact for inner gamma)Method 6 (Gauss−Seidel for all lambdas and exact for inner gamma)Method 7 (Jacobi for all lambdas and exact for inner gamma)

New Challenges

• Stochastic NUM

• Nonconvexity

• Utility function models (energy efficiency, delay, cooperative mode)

• BGP and wireless ad hoc mobile routing

• Complicated ARQ and coding strategies

• Network X-ities

Future Research: Stochastic Issues

• Channel level stochastic

Recent result: stability and optimality proved

• Session level stochastic

Earlier result: Markov model stability for congestion control

Current work: general model stability for cross-layer

• Packet level stochastic

• Topology level stochastic

• Transient behavior characterization

Future Research: Nonconvexity Issues

• Nonconcave utility

(Lee, Mazumdar, and Shroff, Infocom 2004, Chiang, Zhang, and

Hande, Infocom 2005, Fazel and Chiang, CDC 2005)

Sum-of-squares method (Parrilo 2003)

• Nonconvex constraints

Power control in low SIR (Tan, Palomar, and Chiang, Globecom 2005)

Geometric programming and extensions (Chiang 2005)