MASSIMO FRANCESCHETTI

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Wireless sensor networkswith noisy links
MASSIMO FRANCESCHETTI University of California at
Berkeley
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Model of wireless networks
Uniform random distribution of points of density ?
One disc per point
Studies the formation of an unbounded connected
component
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Example
l0.3
l0.4
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Threshold known (only) experimentally
2r
lc(r) 4 p r2 4.5 ENC
ENC is independent of r
Quintanilla, Torquato, Ziff, J. Physics A, 2000
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(No Transcript)
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Maybe the first paper on Wireless Ad Hoc
Networks !
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Ed Gilbert (1961)
P Prob(exists unbounded connected component)
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A nice story
Gilbert (1961)
Physics
Mathematics
Phase Transition Impurity Conduction Ferromagnetis
m Universality (Ken Wilson)
Started the fields of Random Coverage
Processes and Continuum Percolation
Hall (1985) Meester and Roy (1996)
Engineering (only recently)
Gupta and Kumar (1998)
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Engineering
What have we learned from this theory? That
adding more transmitters helps reaching
connectivity so what? (Jan Rabaey)
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Welcome to the real world
Dont think a wireless network is like a bunch
of discs on the plane (David Culler)
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Experiment

• 168 nodes on a 12x14 grid
• grid spacing 2 feet
• open space
• one node transmits Im Alive
• surrounding nodes try to receive message

http//localization.millennium.berkeley.edu
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Connectivity with noisy links
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Unreliable connectivity
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Rotationally asymmetric ranges
How do percolation theory results change?
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Random connection model
Connection probability
Let
define
such that
x1-x2
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Squishing and Squashing
Connection probability
x1-x2
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Example
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Theorem
For all
it is easier to reach connectivity in an
unreliable network
longer links are trading off for the
unreliability of the connection
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Shifting and Squeezing
Connection probability
x
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Example
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Do long edges help percolation?
Mixture of short and long edges
Edges are made all longer
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for the standard connection model (disc)
CNP
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How to find the CNP of a given connection function
Run 7000 experiments
with 100000 randomly placed points in each
experiment
look at largest and second largest cluster of
points (average sliding window 100 experiments)
Assume lc for discs from the literature and
compute the expansion factor to match curves
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How to find the CNP of a given connection function
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Rotationally asymmetric ranges
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Non-circular shapes
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Connectivity
To the engineer as long as ENCgt4.51 we are
fine! To the theoretician can we prove more
theorems?
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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
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Compare three cases
Connection probability
Connection probability
1
1
d
d
Reliable network

• Unreliable network
• independent retransmissions
• dependent retransmissions

ENCunrel ENCrel
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Is shortest path always good?
Not for independent transmissions!
Sink
0.9
0.2
B
Path Hop Count Exp. Num. Trans.
A Sink 1 5
A B Sink 2 2.22
0.9
A
Source
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Shortest path
Max chance of delivery without retransmission
Min expected number of transmissions
Unreliable-dependent
Unreliable-independent
Reliable
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Bottom line
Long links are helpful if you can consistently
exploit them
Connection probability
1
p
x
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Bottom line
Long links are helpful if you can consistently
exploit them
Connection probability
1
p
x
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Acknowledgments
Connectivity L. Booth, J. Bruck, M. Cook.
Routing T. Roosta, A. Woo, D. Culler, S. Sastry