MURI ADCN Workshop

John Doyle, Steven LowEAS, Caltech

OSU, ColumbusSeptember 9, 2009

Post-docsLijun ChenKrister JacobssonChee-Wei Tan

Grad studentsJavad LavaeiJK NairSomayeh Sojoudi

Undergrad/StaffMartin AndreassonTom Quetchenbach

Outline

File fragmentation to mitigate heavy-tailed delay (Low)

Network arch theory (Doyle)

Nonconvex power control in ad hoc wireless networks (Tan)

File fragmentation: summary

Motivation: how to mitigate heavy tail? Recent work showed file transfer time can be

heavy-tailed even if file size is light-tailed(Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007; etc.)

Model Results

Independent or bounded fragmentation preserves light-tailedness

Constant fragmentation min expected delay Asymptotically optimal design: blind

fragmentation Optimal or blind fragmentation preserves tail

index

Model

Given file of random size L L is fragmented into K packets for

transmission at unit rate n-th transmission of size

n-th transmission is successful if

where are iid with distribution F

nx

file fragment constant overhead

nn xA

nA

Model

Ll

xAxll nnnnn

1

1 )( 1

remaining file size at time n+1

fragment size at n

per-packet overhead

iid random var of distr F

Model

Ll

xAxll nnnnn

1

1 )( 1

per-stage cost: )0()( nnn lx 1

total cost:

11

)0( )()(n

nnn

n lxLT 1

Prior work

1

1

1

)0( )()(

)(

nnn

nnnnn

lxLT

Ll

xAxll

1

1

Theorem [Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007]

Without fragmentation T(L) has heavy tail even when L is light-tailed,

provided F has unbounded support

nLxn

Prior work

Theorem [Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007]

Without fragmentation T(L) has heavy tail even when L is light-tailed,

provided F has unbounded support

nLxn

Implication: Heavy tail can originate from protocol interaction alone!

Motivation: How to mitigate?

Prior work

Theorem [Jelenkovic & Tan 2008]

If fragment size = largest of k previous T(L) still has (lighter) heavy tail with first k

moments

nA

1

1

1

)0( )()(

)(

nnn

nnnnn

lxLT

Ll

xAxll

1

1

Prior work

1

1

1

)(

)0()(

)(

nn

nnn

nnnnn

LT

lx

Ll

xAxll

1

1Intuition:

HT is created by repeated comparison of a sequence of iid rv’s with• the same rv L• with unbounded support

,...2,1 ? nLAn

Avoid such fragmentation policies

Two fragmentation policies

independent fragmentation: nnnn XlXx iid ,,min

boundedfragmentation:

nn lbxb ,min

1

1

1

)0( )()(

)(

nnn

nnnnn

lxLT

Ll

xAxll

1

1

Result: LT-preserving frag

independent fragmentation: nnnn XlXx iid ,,min

boundedfragmentation:

TheoremWith independent frag or bounded frag:T(L) is light-tailed provided L is light-tailed

Then, heavy-tailed delay originates only from heavy-tailed files

nn lbxb ,min

What is optimal fragmentation?

)( min : LTxx

n E

Dynamic programming formulation

1

1

1

)0( )()(

)(

nnn

nnnnn

lxLT

Ll

xAxll

1

1

optimalfragmentation:

Per-bit cost

per-bit cost:)(

)(

xFx

xxg

)(minarg0

xgax

x

g(x)

a

Key assumption : is non-decreasing)(

)(

xF

xf

Intuition: a minimizes per-bit cost;Optimal fragment size close to a?

Result: optimal fragmentation

TheoremConstant fragmentation is uniquely optimal

• Optimal #fragments: K*(L) =

• Optimal fragment size: x*(L) = L/K*(L)

•

per-bit cost:)(

)(

xFx

xxg

)(minarg0

xgax

a

Linteger

La

aLx

La

a

/1)(

/1*

Simpler fragmentation

• Optimal fragmentation requires knowledge of L, in addition to failure distr F

• Want: blind fragmentation that only requires F

optimal frag: ,...2,1 ),(* nLxxn

Result: blind fragmentation

Theorem• for all L

• Blind fragmentation is asymptotically optimal

)(minarg0

xgax

LaLx as )(*

blind fragmentation: nn lax ,min

)()()( * aagLJLJ a

expected total cost: )(:)( LTLJ xx E

Result: robustness

Theorem•

•

What happen if the optimal or blind policy is designed wrt failure distribution G when the actual distribution is F ?

)()ˆ()()(1

lim **ˆ agagLJLJL

x

L

Optimal cost under F: )(* LJOptimal cost under G: )(*ˆ LJ x

Blind cost under G: )(ˆ LJ a

)()ˆ()()(1

lim *ˆ agagLJLJL

a

L

Result: tail distribution of T(L)

)(z

DefinitionG is regularly varying(RV) with index a>0 if

where is a slowly varying function

)()( zzzG

0 1)(

)(lim

y

z

zyz

1

1

1

)0( )()(

)(

nnn

nnnnn

lxLT

Ll

xAxll

1

1

Result: tail distribution of T(L)

Theorem• If L light-tailed, so is T(L)

• If L RV(a) (heavy-tailed), so is T(L)

)(~)(

)(~)(*

ag

tLPtLTP

ag

tLPtLTP

a

optimal frag: )(,...,1 ),( ** LKnLxxn

blind frag: nn lax ,min

Result: tail distribution of T(L)

Theorem• If L light-tailed, so is T(L)

• If L RV(a) (heavy-tailed), so is T(L)

)(~)(

)(~)(*

ag

tLPtLTP

ag

tLPtLTP

a

Optimal or blind policy preserves the index of tail distribution

Summary

Independent or bounded fragmentation preserves light-tailedness

Under IFR, optimal fragmentation is unique and constant

Blind fragmentation is asymptotically optimal

Optimal or blind fragmentation preserves tail index

Outline

File fragmentation to mitigate heavy-tailed delay (Low)

Network arch theory (Doyle)

Nonconvex power control in ad hoc wireless networks (Tan)

Network arch theory Key elements of network architecture

Robust yet fragile Layering as optimization

decomposition/distributed IPC Constraints that deconstrain (Gerhart & Kirschner)

ResourcesDeconstrained

ApplicationsDeconstrained

Constraints that deconstrain Xx

pcRx

xUi

iix

)( tosubj

)( max0

Status: very early stage

To better understand layering From familiar: congestion control optimization To: optimal dynamics, wireless, network coding Layering as recursive control: physical layer

antenna design To better understand constraints

Energy constraint Constraints from optimal tradeoffs

Still working on component problems, but optimistic they will point to a general theory

• Each layer is abstracted as an optimization problem

• Operation of a layer is a distributed solution• Results of one problem (layer) are parameters of

others• Operate at different timescales

Xx

pcRx

xUi

iix

)( tosubj

)( max0

Layering as optimization decomposition

Application: utility

IP: routingLink: scheduling

Phy: powerIP

TCP/AQM

Physical

Application

Link/MAC

Layering as optimization decomposition

Network generalized NUM Layers sub-problems Interface functions of primal/dual

variables Layering decomposition methods

• Vertical decomposition: into functional modules of different layers

• Horizontal decomposition: into distributed computation and control

IP

TCP/AQM

Physical

Application

Link/MAC

Examples

application

transport

network

link

physical

Optimal web layer: Zhu, Yu, Doyle ’01

HTTP/TCP: Chang, Liu ’04

TCP: Kelly, Maulloo, Tan ’98, ……

TCP/IP: Wang et al ’05, ……

TCP/power control: Xiao et al ’01, Chiang ’04, ……

TCP/MAC: Chen et al ’05, ……

Rate control/routing/scheduling: Eryilmax et al ’05, Lin et al ’05, Neely, et al ’05, Stolyar ’05, Chen et al ‘06

detailed survey in Proc. of IEEE, 2006

Example: Cross-layer congestion/routing/scheduling design

)}(max))()(( max{min

),()( .. )( max

0

,

:Dual

:Primal

fApxHpxU

ffAxHtsxU

T

f

T

sss

xp

sss

fx

Rate control Scheduling Routing

Rate constraint Schedulability constraint

Cross-layer implementation

Rate control:

Routing: solved with rate control or

scheduling Scheduling:

)()()(maxarg))(()( xHtpxUtpxtx T

sss

x

)()(maxarg))(()( fAtptpftf T

f

Network

Transport

Physical

Application

Link/MAC

A Wi-Fi implementation by Warrier, Le and Rhee shows significantly better performance than the current system.

0min{max ( ) ( )) max ( )} (Dual: T T

s sp x f

s

U x p H x p A f

Rate control Scheduling Routing

Recent generalizationsOptimal control

Lavaei, Doyle and Low, CDC, 2009

Robust control Jacobsson, Andrew and Tang, CDC, 2009 Jacobsson, Andrew, Tang, Low and Hjalmarsson, TAC, March 2009

Game theory Chen, Cui and Low. JSAC, September 2008. Chen, Low and Doyle, ToN, submitted

Network coding Chen, Ho, Chiang, Low and Doyle. T-IT, submitted

Recent generalizationsOptimal control

Lavaei, Doyle and Low, CDC, 2009

Robust control Jacobsson, Andrew and Tang, CDC, 2009 Jacobsson, Andrew, Tang, Low and Hjalmarsson, TAC, March 2009

Game theory Chen, Cui and Low. JSAC, September 2008. Chen, Low and Doyle, ToN, submitted

Network coding Chen, Ho, Chiang, Low and Doyle. T-IT, submitted

ResourcesDeconstrained

ApplicationsDeconstrained

2 2min

arg max , ,

arg max ,s sv

R R dt

L R

x L v

x

v

x c x c

x v p p x c

p

Xx

pcRx

xUi

iix

)( tosubj

)( max0

From optimization to optimal control

router

TCP AQM

my PC

source algorithm (TCP)

iterates on rates

link algorithm (AQM) iterates on prices

cRx

xUs

ssx

s.t.

)( max0

ll

ls l

llssssxp

cppRxxUs

)( max min00

Primal: Dual

horizontal decomposition

Static optimization: dual algorithm

Static optimization: dual algorithm

( , ) ( )

( )

T

T T

L U R

U R

x p x p x c

x p x p c

• Controller is fully decentralized• Globally stable to optimal equilibrium• Generalizations to delays, other controllers

arg max ( , )

R

L

v

p x c

x v p

Implications

arg max ( , )

R

L

v

p x c

x v p

• Views TCP as solving an optimization problem• Clarifies tradeoff at equilibrium• Generalizes to other strategies, other layers• Framework for cross layering

But are the dynamics optimal?

cRxp

TpvL

dtctRxctpxR

v

T

ltx

subject to

))(,( max

||)(||||))((~||2

1 min

0

22

)(

State weigh

t

Control

weight

dynamics

• IQ penalty on deviation from equilibrium• Balance state versus control penalty

arg max ( , )

R

L

v

p x c

x v p

What is this controller optimal for?

Other implications• Elegant proofs of stability• Clarifies the tradeoff in dynamics• Insights about joint congestion control and routing• Can derive more general control laws

))(,(maxarg)(

),(maxarg)(~

subject to

))(,( max

||)(||||))((~||2

1 min

*

0

22

)(

tpvLtx

pvLpx

cRxp

TpvL

dtctRxctpxR

v

v

v

T

ltx

Where we are going

Layering: Rethinking fundamentals• Distributed IPC (Inter-process comms/controls)

– Book: John Day, Patterns in network architecture– Generalizes OS as IPC to networks– Natural fit with optimization framework– Layering/Control recurses, with changes in scope

• Compatible with “platform-based design” (A. S-V)– Recursive design from applications to silicon?– Optimization/decomposition – Illustrate with wireless circuit design

• Emphasis continues on central challenges– Wireless– Mobility– Real time

application

Physical

From layering as DIPC to platform-based design• Recursive design process• From applications to silicon• Optimization/decomposition• Illustrate with antennae

designR

ecursio

n

Sco

pe

Physical

CircuitCircuitCircuit

Logical

Instructions

Next steps

111 )Re( dvv

111 )Im( dvv )Re( dinin yy

)Im( dinin yy

22122122121

1121

211

1121

~)

~~(

~)

~~(

wWZWwWYWWy

ZWYWWV

newTin

newT

0)

~Re()

~Im(

)~

Im()~

Re()~

Re( 11

newnew

newnew

ZZ

ZZW0)

~~Re( 11

1 WZnew

111 )

~~(

~ Tnew YWZ

Antenna

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

ReflectorsReflectors

• Transistors operating at wavelengths << chip dimensions • Forcing (facilitating) integrated E&M, circuits, and systems. • Design difficult but also truly novel systems/capabilities • New and elegant solution for the large-scale radiating circuit problems

where the conventional circuit assumptions are no longer valid (Lavaie, Babakhani, Hajimiri, Doyle)

• Application to diverse wireless communication problems

Unifying theme: Layering as optimizationDuality and convexity

Heterogeneous applications• ubiquitous at every scale• mobility/wireless• real-time/sense/control• exploding complexity and diversity

2 2min

arg max , ,

arg max ,s sv

R R dt

L R

x L v

x

v

x c x c

x v p p x c

p

111 )Re( dvv

111 )Im( dvv )Re( dinin yy

)Im( dinin yy

22122122121

1121

211

1121

~)

~~(

~)

~~(

wWZWwWYWWy

ZWYWWV

newTin

newT

0)

~Re()

~Im(

)~

Im()~

Re()~

Re( 11

newnew

newnew

ZZ

ZZW0)

~~Re( 11

1 WZnew

111 )

~~(

~ Tnew YWZ

Antenna

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

ReflectorsReflectors

Unifying theme: Layering as optimizationDuality and convexity

Outline

File fragmentation to mitigate heavy-tailed delay (Low)

Network arch theory (Doyle)

Nonconvex power control in ad hoc wireless networks (Tan)

Nonconvex Power Control in Ad Hoc Wireless Networks

Chee Wei TanCaltech

Joint Work with Mung Chiang (Princeton) & R. Srikant (UIUC)

45

Motivation

• Objective: Performance Optimization in Multi-hop Ad-hoc Wireless Networks

• Questions:–What are the important performance

objectives in wireless network?– Are there fast algorithms that optimize

the performance objectives?– How to extend the solution to optimize

power and beamformer jointly? 46

Ad Hoc Wireless Networks

47

• Data communication, low power constraint, low complexity signal sets, multiuser interference

Wireless Network Model

• Wireless Ad-hoc Network Model:

•

48

Throughput Maximization

• Total power constraint• Individual power constraint• Vector w as queue length

49

Geometrical Illustration

50University of Illinois at Urbana-Champaign

Two Related Problems

51

Constraints: Individualor total power

Power Control Algorithms

52

• Goal: Fast algorithms under– Weighted Sum Rate maximization– Weighted Max-min SIR– Weighted Sum MSE minimization

– Why? • Time-varying network conditions, i.e., optimization problem

parameters change– Users come and go– Queues of each user change continuously– Due to mobility of users in network– Time-varying fading channel condition

Max-min SIR

53

Interpretation: SIR threshold

Max-min SIR

• Why? - Can express our iterative algorithm as

• Result follows from Blondel, Nivone, Van Dooren (2005), a special case of nonlinear Perron-Frobenius theoy 54

Main result:converges geometrically fast to

right eigenvector of where

where is a nonnegative matrix and is a nonnegative vector.

Weighted Sum MSE

55

Weighted Sum MSE• The problem can be written as

• For a nonnegative matrix where

• Condition: (either low-medium SNR regime or low interference regime)

• Derive using Friedland-Karlin inequalities in nonnegative matrix theory

56

Key Ideas

• Previous approaches: Using geometric programming technique and subgradient technique– Parameter tuning (step-size)– Slow convergence

• Our approach: – Geometric programming change-of-

variable– Show that KKT optimality conditions can

be obtained using a fixed-point approach 57

Weighted Sum MSE: Algorithm

58

Weighted Sum MSE

• Why use this algorithm?– Geometrically fast convergence– No step-size tuning required

59

• Proof outline:– z = I (z)– Under conditions on I(.), convergence is

geometric, results followed from Yates (1995)

– Our MSE algorithm can be shown to satisfy these conditions

Weighted Sum Rate

60

Weighted Sum Rate• The problem can be written as

• For a nonnegative matrix where

• Same idea as Weighted Sum MSE problem• KKT optimality conditions can be obtained using a

fixed-point approach

61

Weighted Sum Rate: Algorithm

62

• In general, Max-min SIR not the same as Weighted Sum Rate

63

Connection between Weighted Sum Rate & Weighted Max-min

SIR

User 1 rate

User 2 rate

Vector w (queue size)

• In general, Max-min SIR not the same as Weighted Sum Rate

64

Connection between Weighted Sum Rate & Weighted Max-min

SIR

Vector w (queue size)

User 1 rate

User 2 rate

Connection between Weighted Sum Rate & Weighted Max-min SIR

65

Extensions

• So far work for ad hoc networks or single-antenna power controlled networks

• For MIMO networks, need to optimize beamformers

• Initial work: Access-point controlled network

66

Downlink Transmit Beamformer

• Optimize power and transmit beamformer for all users

• Goal: Max-min SIR over power and beamformers

67

Transmitbeamformer

u1

u2

User 1

User 2

User 1Receiver

User 2Receiver

Power control

Uplink Receive Beamformer

68

Receivebeamformer

u1

u2

User 1

User 2

User 1Transmitter

User 2Transmitter

Power control

• Virtual uplink as auxiliary mechanism• Our approach: Iterative solution is easier, reuses

existing CDMA power control module and converges geometrically fast

timeSlot 1(Downlink)

Slot 2(Uplink)

Slot 3(Downlink)

….