Robust Mulilayer Design of Wireless Networks for Distributed Systems

Andrea GoldsmithStanford University

wsl.stanford.edu

IPAM WorkshopMay 16, 2002

Massively Distributed Systems

Challenges

Design and capacity of large wireless adhoc networks are open problems

Hard energy and delay constraints change fundamental design principles

Many applications fail miserably with a “generic” network approach

Example: String Stability

Applied to vehicle platoons with linear controllersString stable if spacing error decreases along platoon

Communication system: token passing WLAN

Controllers unstable for any delay in lead vehicle information

Lead vehicle broadcasts or controller redesign stabilizes the system under bounded delay

Multilayer Design Hardware

Power or hard energy constraints Size constraints

Link Design Time-varying low capacity channel

Multiple Access Resource allocation (power, rate, BW) Interference management

Networking. Routing, prioritization, and congestion control

Application Real time media and QOS support Hard delay/quality constraints

Multilayer Design

Design Issues

Some applications require tight coupling across layers, while others can be more flexible

Diversity and adaptability are essential for robustness

What information should be exchanged across layers and how it should be used

Outline

Fundamental capacity limits Adaptive modulation and resource

allocation Medium access control Ad hoc network design Energy constrained networks Joint control and communication design Multilayer network design

Wireless Channel Capacity

Fundamental Limit on Data Rates

Main drivers of channel capacityBandwidth and powerStatistics of the channelChannel knowledge at transmitter/receiverNumber of antennasMinimum rate and delay constraints

Capacity: The set of simultaneously achievable rates {R1,…,Rn}

R1R2

R3

Broadcast: One Transmitter to Many Receivers.

Multiple Access: Many Transmitters to One Receiver.

Min-Rate Capacity Region:

Severe Rician Fading

Independent Rician fading with K=1 for both users(severe fading, but not as bad as Rayleigh).

P = 10 mW, B = 100 KHz

Adaptive Modulation and Coding in Flat

Fading

Adapt transmission to channelParameters: power,rate,code,BER, etc.Capacity-achieving strategy

Recent WorkAdaptive modulation for voice and data (to meet QOS)Adaptive turbo coded modulation (<1 db from capacity)Multiple degrees of freedom (only need exploit 1-2)Adaptive power, rate, and compression with hard deadlines

(t)

UncodedData Bits Buffer

M()-QAM ModulatorPower: S()

To Channel(t)

PointSelector

log2 M() Bits One of theM() Points

BSPK 4-QAM 16-QAM

Adaptation under Hard

Delay Constraints

30ms constraint

90ms constraint

Optimal Power Control and Joint Source/Channel Coding

Data Rate (bps)

Pow

er (

mW

)

Ad-Hoc Network Capacity

Each node generates independent data. Source-destination pairs are chosen at random. Routing can be multihop. Topology is dynamic Generally a fully connected network with different link

SNRs Can allocate resources dynamically (rate, power, BW,

routes,…)

Capacity RegionAll achievable rate vectors between

nodesAn n(n-1) dimensional convex

polyhedronEach dimension defines (net) rate from one

node to each of the others

AchievabilityTime divisionAWGN or flat fadingCentralized control

Converse

3

1

2

4

5

Rate Matrix

1 2

3 4

Data from 1, rate 10

Data from 2, rate 20

0000

0000

202000

001010

R

Transmission Scheme Rate Matrix

Transmission scheme at time t for n users (snapshot)Rows represent original data sourceNegative entries represent bits to send or forwardPositive entries represent bits received (data rate)Link rates dictated by link capacity given SIR

(variable rate)Multihop routing and power control increase set of

matrices

Time Division

Time division of two schemes is a linear combination of their rate matrices.

Example: 50% of time under scheme A and 50% of time under scheme B has rate matrix:

5500

0000

1010200

0055

101000

0000

040400

0000

5.0

0000

0000

202000

001010

5.0

Scheme A Scheme B 50/50 Time Division

User 1 sends 5 bps/Hz to User 2User 2 sends 10 bps/Hz to User 3 and 10 bps/Hz to User 4User 4 sends 5 bps/Hz to User 3

Capacity Region

A matrix R belongs to the capacity region if there are rate matrices R1, R2, R3 ,…, Rn such that

Linear programming problem: Need clever techniques to reduce complexityPower control, fading, etc., easily incorporatedRegion boundary achieved with optimal routing

0 , 1 ,1 1

i

n

ii i

n

iia a R a R

Achievable ratevectors achieved by time division

Capacity region is convex hull ofall rate matrices

Example: Six Node Network

Capacity region is 30-dimensional

Capacity Regions(a): Single hop, no simultaneous transmissions.(b): Multihop, no simultaneous transmissions. (c): Multihop, simultaneous transmissions.(d): Adding power control (e): Successive interference cancellation, no power control.

jiijRij ,34,12 ,0

Multiplehops

Spatial reuse

SIC

Extensions: - Capacity vs. network size - Energy constraints - Fading and mobility - Multihop cellular

Fading increases capacity

(a): No routing, no simultaneous transmissions.(b): Routing, no simultaneous transmissions.(c): Routing, simultaneous transmissions.(d): Adding power control.(e): Successive interference cancellation, no power control.

Gain matrix alternates between N fading states

In a similar way, mobility also increases capacity

Shannon Capacity of Ad-Hoc Networks

For n nodes, p(x(1),…,x(n)) s.t. (Cover/Thomas)

},...,1{),|;( )()()(

,

nSXYXIRcc

c

SSS

SjXiij

Rate flow across cutsets bounded by conditional MI

S ScXi

Yj

Yk

Xk

Relay transmissions

Nodes can transmit directly and/or use other nodes as relays

Relay Channel Results Direct plus one relay (Cover,El Gamal’79)

Parallel relays (Schein,Gallager’00)

Source Destination

N1+

+

N2

SourceDestination

N1+

+

N2

+

N3

Capacity Strategy:- Broadcast coding-Cooperative MAC coding- Source coding-Random, list, block Markov codes

Capacity Upper Bounds1) Data processing thm2) Cover/El Gamal result

Achievability1) Staggered block coding2) Transponder schemeBounds not tight: hard problem

Capacity Ideas for Ad Hoc Networks

Multiple Antenna (MIMO) ChannelsCan obtain large capacity increases with multiple

antennas In sensor networks, sensor clusters can utilize

these gains

Interference “Dirty paper” coding removes the effect of known

interference without increasing required transmit power

Shannon capacity ignores data arrival statistics

Does MAC capacity change for bursty data?Can only decrease

Need better transmission strategies for Aloha

Need better methods of collision resolution

Random Access

Medium Access Control

Nodes need a protocol for channel accessMinimize packet collisions and insure channel not

wastedCollisions entail significant delay

First protocols designed for fully-connected networksSuffer from hidden and exposed terminal problems

802.11 uses four-way handshakeCreates inefficiencies, especially in multihop setting

HiddenTerminal

ExposedTerminal

1 2 3 4 5

Multiple mini-slots

Multiple mini-slots increase efficiency of collision resolution

Different minislot protocols investigated Distributed p-Persistent Algorithm (DPA) Distributed Splitting Algorithm (DSA) Distributed Token Bus (DTB)

Propagation delay factored in guard times Non FIFO queueing also improves efficiency

Time

mini-slot pairsdata slot

Throughput versus Delay

(a): Theoretical bound

(b): IEEE802.11 upper bound

(c): IEEE802.11

(d): DPA

(d’): non-FIFO DPA

(e): DSA

(e’): non-FIFO DSA

(f): DPA

(f’): non-FIFO DPA

(a)

(b)

(d)

(c)

(e’)

(d’)

(f’)

(f)

(e)

Numerical results obtained via discrete event simulation

DTB Capacity Region

(a): Theoretical bound

(b): IEEE802.11 upper bound

(c): IEEE802.11

(d): DPA

(e): DSA

(f): DPA(d)

(b)

(c)

(a)

(e)

(f)

MAC with Data Prioritization

Each user transmits whenever he has data to send

Coding strategy: Combine broadcast and MACEach user sends a multiresolution signalWithout collisions all data gets throughWith collisions some data gets through

Lost bits may be retransmitted

2=p2L/T

1=p1L/T

Results High priority data always gets through

This coding strategy achieves capacity If (1,2)C, these rates will be achievedBurstiness does not decrease capacity!

Superposition coding only needed when users have very different SNRsOtherwise code for constant collisions or no collisions,

depending on pi.

Show that queues in system are stable for any rate pair (1,2) inside MAC capacity region.

Networks with Energy-Constrained Nodes

Capacity per unit energy (Gallager’87, Verdu’90)Number of bits per unit energy such that error

probability decreases to zero with increasing energy

Not possible to send a finite number of bits with finite energy and Pe arbitrarily small

Energy per bit minimized by sending bits over many dimensions (symbols,time,BW)New communication system paradigm

Network designs must now consider node lifetime (among other things) in MAC and routing protocols

Energy Constrained Networks

Channel capacity is the maximum possible rate with arbitrarily small Pe (reliable transmission) Input often has an average or peak power constraint

Capacity per unit cost (Gallager’87, Verdu’90)Number of bits that can be transmitted per unit cost

for sending these bits (cost is typically energy)

Not possible to send a finite number of bits with finite energy and Pe arbitrarily small

Capacity per unit energy achieved with on-off signalling

We investigate dynamic rate, power, and routing strategies for networks with finite-energy nodes

Bits per Unit Energy General channels with a “0” (Verdu’90)

Gaussian channels with energy E and M messages

Minimum energy per bit:

Codes arbitrarily long for small Pe, and E

Exx

ppDC

n

i

XYxXYx

1

20|| ,2

)||(sup~

2/1log

2

log~

0

2

nN

E

E

n

E

MC

2ln)5./1log(5.

/lim~

1limmin 0

0

NnNE

nE

CE

nnb

Energy vs. Symbol per Bit

Energy/bit

Symbols/bit

N0 ln2

Minimum energy per bit achieved with many degrees of freedom

Can fading help? For most fading distributions, channel gain is

large with small probability

With finite energy, can transmit any number of bits with Pe arbitrarily smallTransmit when channel is “good”Delay can be largeCapacity per unit energy typically infinite

We consider maximizing the number of bits transmitted reliably over a block fading channelDelay constraint: can’t average over all fading values

System Model m blocks of n symbols (n large)

m represents delay constraintEach block has small but nonzero Pe

Fading gain on ith block is g[i] (i.i.d.)Transmitter and receiver know g[i] at time i

Energy on ith block:

Effective energy on ith block:

X11,…,X1n X21,…,X2n Xm1,…,Xmn

g1 g2 gm

n

k ikXiE1

2][

n

kikeff XigiEigiE

1

2][][][][

Maximizing Transmitted Bits

AWGN channel with gain g and energy E:Minimum energy per bit: N0 ln 2/gBits per unit energy: g/(N0 ln 2)Total number of bits sent: B=gE/(N0

ln 2)

For block fading, bits sent in ith frame:

m

i

EiEN

iEigiB

10

][,2ln

][][][

Goal: optimally allocate E to maximize sum of bits

Problem Formulation

Optimizing Energy Allocation

Finite horizon dynamic programmingValue iteration algorithm

m

i

m

imggmEE

EiEN

iEigEB

11 0][],...,1[][],...,1[

* ][,2ln

][][max

miigJEN

iEigigJ ii

,])]1[([,

2ln

][][max])[( 1

0

2ln

][][])[(

0N

mEmgmgJm

Threshold Policy

Energy allocated according to threshold rule

Recursion for i:

Threshold decreases with each block: ii+1

])]1[([2ln

,0

][][ 1

0

igJEE

N

else

igEiE ii

i

Use all energy in current block if fading exceeds expected future gains

1

)()(,)()( 11

i

xgpxPdxxxpP iii

Threshold LevelThreshold Level in Rayleigh Fading for m=20

Th

resh

old

i

Block Number 0 2 4 6 16 18 20 14 12 10 8

0

0.5

1

1.5

2

2.5

3

3.5

Transmit

Don’t Transmit

Capacity Evaluation

Maximum number of transmitted bits

dxxpN

xEB

m

ii

)(2ln1 0

*

Capacity in Rayleigh Fading

Block Number 0 10 20 30 80 90 100 70 60 50 40

0

5

10

15

Energy Constrained Routing

Ad hoc network with n nodesLink gains between nodes are Gij.Each node has finite energy Ei

Minimum energy to send 1 bit on link ij is N0ln2/Gij

Maximize the total number of bits sent from A0 to An-1 given the node energy constraints

A0

An-1

Minimum Energy Routing

Routing strategy for each bit:Choose a route from A0 to An-1 with the

minimum total energy per bit (minimum cost)

Shortest path problemSolved using dynamic programmingReduce node energy after each transmissionTotal number of transmitted bits depends on

node energies

ij

Tb

AAA G

NE

n

2lnminarg 0

),...,,{

*

110

Joint Control and Network Design

Robust controllers compensate for modeling errorsThere is little known about incorporating random

packet delays and losses into controller design

Network-robust controllers must compensate for asynchronous, delayed, and lossy information

Network tradeoffs impact controller performanceRate vs delay, hard deadlines, energy constraints.

Network requirements defined by controller design

Network and controller should be jointly designed

Fundamental Trade-offs

Effects of communication faults on controller

High data rates, low latency, low packet loss are competing objectives in wireless networks.

Control system

Data rates Quantization noise

Random Packet Delay Delay and asynchronicity in feedback

Packet loss Vacant sampling

General Problem Setup

LTIPlant

RemoteController

Sh Hh

WirelessLink

WirelessLink

Noise & disturbance

Measured outputs

Sampled outputs

Regulated outputs

Desired control input

Actual control input

Goal: Investigate effects of quantization noise, packet loss/delay,and link design and adaptation on the controller performance.

Performance We consider both hard and soft decoding on the link

Soft decision implies no packet errors or lossHard decision entails random packet loss

H2 norm – the covariance in the regulated output when the driven noise is N(0, I).

Hybrid system (sampled-data system) is not LTI, but it is periodic. We use a generalized H2 norm.

Sampled-data H2 optimal control solved via an associated discrete-time system, which depends on sample period h.

With packet loss, we use the covariance in the regulated output as performance measure. The regulated output is a Gaussian mixture Its statistics are time-varying.

Robustness to Packet Loss

Robustness to Imperfect Communication

Performance Comparison(average power = .01)

Multilayer Design Issues

Network VariationsVariations at take place on difference timescalesVariations should be adapted to locally and globally

Fundamental QuestionsWhat information should be exchanged across

layers?How should that information be used at each layer?Where do “separation theorems” apply? If guaranteed QoS not possible, then what?

CoordinationHow to balance the needs of all users/applications

Conclusions Multilayer design of networks an open

problem

Energy and delay constraints require new design philosophies

Some applications require joint design of hardware, link, network, and application protocols.