MASSIMO FRANCESCHETTI University of California at Berkeley Wireless sensor networks with noisy links

Continuum percolation theory Meester and Roy, Cambridge University Press (1996) Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component

Model of wireless networks Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component A B

 0.3  0.4 Example

[Quintanilla, Torquato, Ziff, J. Physics A, 2000] c (r) 4  r 2 = 4.5 = ENC 2r Threshold known (only) experimentally ENC is independent of r   c (1) r2r2 r2r2 == 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) λcλc λ2λ2 1 0 λ P λ1λ1 P = Prob(exists unbounded connected component)

A nice story Gilbert (1961) Mathematics Physics Started the fields of Random Coverage Processes and Continuum Percolation Engineering (only recently) Gupta and Kumar (1998) Phase Transition Impurity Conduction Ferromagnetism Universality (…Ken Wilson) Hall (1985) Meester and Roy (1996)

Engineering “What have we learned from this theory? That adding more transmitters helps 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

Prob(correct reception) Connectivity with noisy links

Unreliable connectivity 1 Connection probability d Continuum percolation 2r Random connection model d 1 Connection probability

Rotationally asymmetric ranges How do percolation theory results change?

Random connection model Connection probability ||x 1 -x 2 || define Let such that

Squishing and Squashing Connection probability ||x 1 -x 2 ||

Connection probability 1 ||x|| Example

Theorem 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 Connection probability ||x||

Example Connection probability ||x|| 1

Mixture of short and long edges Edges are made all longer Do long edges help percolation?

CNP Squishing and squashing Shifting and squeezing for the standard connection model (disc)

c =  How to find the CNP of a given connection function Run 7000 experiments with randomly placed points in each experiment look at largest and second largest cluster of points (average sliding window 100 experiments) Assume c for discs from the literature and compute the expansion factor to match curves

How to find the CNP of a given connection function

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, but how 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 Connection probability d d 1 Connection probability Reliable network Unreliable network independent retransmissions dependent retransmissions ENC unrel = ENC rel

Is shortest path always good? Source A B Sink PathHop Count Exp. Num. Trans. A Sink15 A B Sink22.22 Not for independent transmissions!

Max chance of delivery without retransmission Shortest path Min expected number of transmissions Unreliable-dependent Reliable Unreliable-independent

Bottom line Long links are helpful if you can consistently exploit them Connection probability 1 ||x|| p

Bottom line Long links are helpful if you can consistently exploit them Connection probability 1 ||x|| p N hops vs. N hops (no retransmission) N hops vs. hops (with indep. retransmission)

Acknowledgments Connectivity: L. Booth, J. Bruck, M. Cook. Routing: T. Roosta, A. Woo, D. Culler, S. Sastry