MASSIMO FRANCESCHETTIUniversity of California at Berkeley

Wireless sensor networkswith noisy links

Continuum percolation theoryMeester and Roy, Cambridge University Press (1996)

Uniform random distribution of points of density λ

One disc per pointStudies the formation of an unbounded connected component

Model of wireless networks

Uniform random distribution of points of density λ

One disc per pointStudies the formation of an unbounded connected component

A

B

0.3 0.4

Example

[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

c(r) 4 r2 = 4.5 = ENC

2r

Threshold known (only) experimentally

ENC is independent of r

0.35910c(1)

r2 r2==

c(r)

Maybe the first paper on Wireless Ad Hoc Networks !

Theory

To model wireless multi-hop networks

Ed Gilbert (1961)(following Erdös and Rényi)

Ed Gilbert (1961)

λc λ2

1

0

λ

P

λ1

P = Prob(exists unbounded connected component)

A nice story

Gilbert (1961)

Mathematics Physics

Started the fields ofRandom Coverage Processesand Continuum Percolation

Engineering (only recently)Gupta and Kumar (1998)

Phase TransitionImpurity Conduction

FerromagnetismUniversality (…Ken Wilson)

Hall (1985)Meester and Roy (1996)

Engineering

“What have we learned from this theory? That adding more transmittershelps reaching connectivity…

…so what?”

(Jan Rabaey)

Welcome to the real world

“Don’t think a wireless network is like a bunch of discs on the plane” (David Culler)

•168 nodes on a 12x14 grid• grid spacing 2 feet• open space• one node transmits “I’m Alive”• surrounding nodes try to receive message

Experiment

http://localization.millennium.berkeley.edu

Prob(correct reception)

Connectivity with noisy links

Unreliable connectivity

1

Connectionprobability

d

Continuum percolationContinuum percolation

2r

Random connection modelRandom connection model

d

1

Connectionprobability

Rotationally asymmetric ranges

How do percolation theory results change?

Random connection model

Connectionprobability

||x1-x2||

)( 21 xxg

2

)())((0x

xgxgENC

]1,0[:)( 221 xxgdefine

Let 221, xx

such that

Squishing and Squashing

Connectionprobability

||x1-x2||

))(()( 2121 xxpgpxxgs

)( 21 xxg

2

)())((x

xgxgENC

))(())(( xgsENCxgENC

Connectionprobability

1

||x||

Example

2

)(0x

xg

Theorem

))(())(( xgsxg cc

For all

“longer links are trading off for the unreliability of the connection”

“it is easier to reach connectivity in an unreliable network”

Shifting and Squeezing

Connectionprobability

||x||

)(

0

1

)()(

))(()(yhs

s

y

dxxxgxdxxgss

xhgxgss

)(xg

2

)())((x

xgxgENC

))(())(( xgssENCxgENC

)(xgss

Example

Connectionprobability

||x||

1

Mixture of short and long edges

Edges are made all longer

Do long edges help percolation?

2

51.44)(

...359.0

2

2

rdxxgCNP

r

cc

c

CNP

Squishing and squashing Shifting and squeezing

for the standard connection model (disc)

Prob(Correct reception)

Rotationally asymmetric ranges

CNP

Is the disc the hardest shape to percolate overall?

Non-circular shapes

CNP

To the engineer: as long as ENC>4.51 we are fine!To the theoretician: can we prove more theorems?

Connectivity

The network is connected, buthow do I get packets to destination?

Two extreme cases:

• Re-transmissions are independent (channel is highly variant)

• Re-transmissions have same outcome (channel is not variant)

Flip a coin at every transmission

Flip a coin only once to determine network connectivity

Compare three cases

1

Connectionprobability

d d

1

Connectionprobability

Reliable network Unreliable network• independent retransmissions• dependent retransmissions

ENCunrel= ENCrel

Is shortest path always good?

0.9

0.9

0.2

SourceA

B

Sink

Path Hop

Count

Exp. Num. Trans.

A Sink 1 5

A B Sink 2 2.22

Not for independent transmissions!

Max chance of delivery without retransmission

Shortest path

Min expected number of transmissions

Unreliable-dependent

Reliable

Unreliable-independent

Bottom lineLong links are helpful if you can consistently exploit them

Connectionprobability

1

||x||

p

p

RR

Bottom lineLong links are helpful if you can consistently exploit them

Connectionprobability

1

||x||

p

p

RR

N hops vs. N hops (no retransmission)

N hops vs. hops (with indep. retransmission)

p

p

N

Acknowledgments

Connectivity: L. Booth, J. Bruck, M. Cook.

Routing: T. Roosta, A. Woo, D. Culler, S. Sastry