Robust Mulilayer Design of Wireless
Networks for Distributed Systems
Andrea Goldsmith
Stanford University
wsl.stanford.edu
IPAM Workshop
May 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 controllers

String 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



Link Design


Resource allocation (power, rate, BW)
Interference management
Networking.


Time-varying low capacity channel
Multiple Access



Power or hard energy constraints
Size constraints
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
Capacity: The set of simultaneously achievable rates {R1,…,Rn}
R1
R2
R3
Broadcast:
One Transmitter
to Many Receivers.
Multiple Access:
Many Transmitters
to One Receiver.

Main drivers of channel capacity





Bandwidth and power
Statistics of the channel
Channel knowledge at transmitter/receiver
Number of antennas
Minimum rate and delay constraints
Min-Rate Capacity Region:
Severe Rician Fading
P = 10 mW,
B = 100 KHz
Independent Rician fading with K=1 for both users
(severe fading, but not as bad as Rayleigh).
Adaptive Modulation
and Coding in Flat Fading
Uncoded
Data Bits
Point
Selector
Buffer
log2 M(g) Bits
g(t)

g(t)
To Channel
Adapt transmission to channel



One of the
M(g) Points
M(g)-QAM
Modulator
Power: S(g)
Parameters: power,rate,code,BER, etc.
Capacity-achieving strategy
Recent Work




BSPK
4-QAM
16-QAM
Adaptive 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
Adaptation under Hard
Delay Constraints
Power (mW)
 Optimal Power Control and Joint Source/Channel Coding
30ms constraint
90ms constraint
Data Rate (bps)
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 Region

All achievable rate vectors between nodes

An n(n-1) dimensional convex polyhedron
 Each
dimension defines (net) rate from one
node to each of the others

Achievability
 Time division
 AWGN or flat fading
1
 Centralized control

Converse
3
2
5
4
Rate Matrix

Transmission scheme at time t for n users (snapshot)
Rows represent original data source
Negative entries represent bits to send or forward
 Positive 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
Transmission Scheme
Rate Matrix


1
Data from 1, rate 10
3
Data from 2, rate 20
2
4
0
0
 10 10
 0

0

20
20

R
 0
0
0
0


0
0
0
0


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:
0
0
0
0
0   5
5
0 0
 10 10
0
 0

0  40 40 0   0  20 10 10 
0

20
20
  0.5  


0.5  
 0
0
0
0
0
0
0
0  0
0
0 0



 

0
0
0
0
10  10  0
0
5  5
 0
0
Scheme A
Scheme B
50/50 Time Division
User 1 sends 5 bps/Hz to User 2
User 2 sends 10 bps/Hz to User 3 and 10 bps/Hz to User 4
User 4 sends 5 bps/Hz to User 3
Capacity Region
Achievable rate
vectors achieved
by time division

A matrix R belongs to the capacity region if there are rate
matrices R1, R2, R3 ,…, Rn such that
R   ai Ri ,
i 1
n

Capacity region
is convex hull of
all rate matrices
a
i 1
i
 1,
ai  0
n
Linear programming problem:
 Need clever techniques to reduce complexity
 Power control, fading, etc., easily incorporated
 Region boundary achieved with optimal routing
Example: Six Node Network
Capacity region is 30-dimensional
Capacity Regions
Rij  0, ij  12,34, i  j
Multiple
hops
Spatial
reuse
SIC
(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.
Extensions:
- Capacity vs. network size
- Energy constraints
- Fading and mobility
- Multihop cellular
Fading increases capacity

Gain matrix alternates between N fading states
(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.

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)
R
ij
 I(X
(S)
;Y
(Sc )
|X
(Sc )
iX , jS c
S  {1,..., n}
),
Relay transmissions
S
Sc
Xi
Yk
Xk
Yj
Rate flow across cutsets bounded by conditional MI
Nodes can transmit directly and/or use other nodes as relays
Relay Channel Results

Direct plus one relay (Cover,El Gamal’79)
N2
Source
Destination
+
+
N1

Capacity Strategy:
- Broadcast coding
-Cooperative MAC coding
- Source coding
-Random, list, block Markov codes
Parallel relays (Schein,Gallager’00)
N3
+
N2
Source
Destination
+
N1
+
Bounds not tight: hard problem
Capacity Upper Bounds
1) Data processing thm
2) Cover/El Gamal result
Achievability
1) Staggered block coding
2) Transponder scheme
Capacity Ideas for
Ad Hoc Networks
 Multiple
Antenna (MIMO) Channels
Can 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
Random Access

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
Medium Access Control

Nodes need a protocol for channel access



Minimize packet collisions and insure channel not wasted
Collisions entail significant delay
First protocols designed for fully-connected networks

Suffer from hidden and exposed terminal problems
Hidden
Terminal
Exposed
Terminal
1

2
3
4
5
802.11 uses four-way handshake

Creates inefficiencies, especially in multihop setting
Multiple mini-slots
mini-slot pairs
data slot
Time


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
Throughput versus Delay
(f)
(b)
(a)
(e)
(c)
(d’)
(a): Theoretical bound
(b): IEEE802.11 upper
bound
(d)
(f ’)
(c): IEEE802.11
(d): DPA
(d’): non-FIFO DPA
(e): DSA
(e’)
(e’): non-FIFO DSA
(f): DPA
(f ’): non-FIFO DPA
Numerical results obtained via discrete event simulation
DTB Capacity Region
(a): Theoretical bound
(f)
(a)
(b): IEEE802.11 upper
bound
(c): IEEE802.11
(e)
(d): DPA
(b)
(c)
(d)
(e): DSA
(f): DPA
MAC with Data Prioritization
l2=p2L/T
l1=p1L/T

Each user transmits whenever he has data to send

Coding strategy: Combine broadcast and MAC
Each user sends a multiresolution signal
 Without collisions all data gets through
 With collisions some data gets through


Lost bits may be retransmitted
Results

High priority data always gets through

This coding strategy achieves capacity



Superposition coding only needed when users
have very different SNRs


If (l1,l2)C, these rates will be achieved
Burstiness does not decrease capacity!
Otherwise code for constant collisions or no collisions,
depending on pi.
Show that queues in system are stable for any rate
pair (l1,l2) inside MAC capacity region.
Networks with EnergyConstrained 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)
~ sup x D( pY | X  x || pY | X 0 )
C
,
2
x

n
2
x
 E
i 1
Gaussian channels with energy E and M messages

n
E 
~ log 2 M

C

log 1 
E
2E
 nN 0 / 2 

Minimum energy per bit:
1
E/n
min Eb  lim ~  lim 
 N 0 ln 2
n  C
n 
.5 log( 1  E / .5nN 0 )

Codes arbitrarily long for small Pe, and E
Energy vs. Symbol per Bit
Energy/bit
N0 ln2
Symbols/bit
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 small




Transmit when channel is “good”
Delay can be large
Capacity per unit energy typically infinite
We consider maximizing the number of bits
transmitted reliably over a block fading channel

Delay constraint: can’t average over all fading values
System Model

m blocks of n symbols (n large)



m represents delay constraint
Each 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
X11,…,X1n
g1
X21,…,X2n
g2
Xm1,…,Xmn
gm

Energy on ith block: E[i ]  k 1 X ik2

Effective energy on ith block:
n
n
E eff [i ]  g[i ]E[i ]  g[i ] X ik2
k 1
Maximizing Transmitted Bits

AWGN channel with gain g and energy E:
 Minimum energy per bit: N0 ln 2/g
 Bits 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:
g[i ]E[i ]
B[i ] 
,
N 0 ln 2
m
 E[i ]  E
i 1
Goal: optimally allocate E to maximize sum of bits
Problem Formulation

Optimizing Energy Allocation
B*  max
E [1],..., E [ m ]

 m g[i ]E[i ] 
E 
,

g[1],..., g[ m ]
 i 1 N 0 ln 2 
m
 E[ i ]  E
i 1
Finite horizon dynamic programming
 Value
iteration algorithm
 g[i ]E[i ]

J i ( g[i ])  max 
, E[ J i 1 ( g[i  1])] ,
 N 0 ln 2

J m ( g[m ]) 
g[m ]E[m ]
N 0 ln 2
im
Threshold Policy

Energy allocated according to threshold rule
E
E[ i ]  
0
g[i ]  a i
,
else
N 0 ln 2
ai 
E[ J i 1 ( g[i  1])]
E
Use all energy in current block if fading exceeds expected future gains

Recursion for ai:
a i  P (a i 1 )a i 1 

a xp( x)dx,
P ( x )  p( g  x )
i 1

Threshold decreases with each block: aiai+1
Threshold Level
Threshold Level in Rayleigh Fading for m=20
3.5
3
Transmit
Threshold ai
2.5
2
1.5
Don’t Transmit
1
0.5
0
0
2
4
6
8
10
Block Number
12
14
16
18
20
Capacity Evaluation

Maximum number of transmitted bits
m 
xE
B 
p( x )dx
i 1 a i N 0 ln 2
*
Capacity in Rayleigh Fading
15
10
5
0
0
10
20
30
40
50
Block Number
60
70
80
90
100
Energy Constrained Routing
A0

Ad hoc network with n nodes




An-1
Link 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
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)
 
*

arg min
 { A0 , A1 ,..., An1 )
 T
N 0 ln 2 
 Eb  

Gij 


Shortest path problem



Solved using dynamic programming
Reduce node energy after each transmission
Total number of transmitted bits depends on node energies
Joint Control and
Network Design

Robust controllers compensate for modeling errors

There 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 performance


Rate 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
Control system
Data rates
Random Packet Delay
Packet loss

Quantization noise
Delay and asynchronicity in feedback
Vacant sampling
High data rates, low latency, low packet loss are
competing objectives in wireless networks.
General Problem Setup
Regulated
outputs
LTI
Plant
Sh
Sampled
outputs
Wireless
Link
Noise &
disturbance
Measured
outputs
Hh
Actual control
input
Remote
Controller
Desired
control input
Wireless
Link
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 loss
 Hard 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 Variations



Fundamental Questions





Variations at take place on difference timescales
Variations should be adapted to locally and globally
What 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?
Coordination

How 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.