Post on 20-Dec-2015
transcript
Where are we heading
year
Lo
g (
peop
le p
er
com
pu
ter)
Number CrunchingData Storage
NetworkingCommunication
Interaction with the physical world
Building Comfort,Smart Alarms
Great Duck Island
Elder Care
Fire Response
Factories
Wind ResponseOf Golden Gate Bridge
Vineyards
Redwoods
Instrumenting the world
Soil monitoring
• Connected ?
• Information throughput ?
• Transmission power ?
• routing ?
• delay ?
• reliability ?
Extreme scaling
Large scale networks theory
• Spatial stochastic networks
• Connectivity
• Sensor placement algorithms
• Throughput capacity
• Closing the loop
• Physics of propagation
• Distributed Algorithms
Talk Outline
Poisson distribution of points of density λ
Points are connected if their distance is less than 2rStudies the formation of connected components
Continuum percolation
S
D
Gilbert J.SIAM (1961)
Book: Meester & Roy (1995)
λc λ2
1
0
λ
P
λ1
P = Prob(exists unbounded connected component)
Phase transition Theorem Gilbert J.SIAM (1961)
Networks with interference
Communication range is different from connectivity
Nodes in close range might not be connected
Dependent percolation model
What happens to the phase transition?
A percolation model for wireless networks
Node j can receive data from node i if the signal to noise plus interference ratio is above a threshold
All nodes transmit at power P
A percolation model for wireless networks
Gupta Kumar, IEEE Trans. IT, 2000]
Gilbert, J. SIAM, 1961]
[Dousse Baccelli Thiran, IEEE Trans. Net, 2004]
[Dousse Franceschetti Meester Thiran, preprint, 2004]
10
All nodes transmit at power PNode j can receive data from node i if the signal to noise plus interference ratio is above a threshold
Can we prove a stronger result ?
c
super-c
ritica
l
sub-cr
itical
No interference model (Gilbert)
Interference model
super-c
ritica
l
Theorem
c
super-c
ritica
l
sub-cr
itical
(Dousse Franceschetti Meester Thiran)
No interference model (Gilbert)
Interference model
super-c
ritica
l
Bottom line
An ideal network with perfect interference cancellation (independent percolation model) exhibits a phase transition for c
A network where nodes cause interference (dependent percolation model) exhibits a phase transition for c,
More work on connectivity
• Gilbert’s model
• Interference model
• Spread out, unreliable connections
Prob(correct reception)
Let’s look at some real data
•168 nodes on a 12x14 grid• grid spacing 2 feet• only one node transmits “I’m Alive” (no interference)• surrounding nodes try to receive message
http://localization.millennium.berkeley.edu
Absence of sharp threshold
1
Connectionprobability
||xi-xj ||||xi-xj||
1
Connectionprobability
How does the critical density change with the shape of the connection
probability?
c
longer links are trading off for the unreliability of the connection
TheoremFranceschetti Booth Cook Bruck Meester (2003)
It is easier to reach connectivity with unreliable spread out connections
More work on connectivity
• Gilbert’s model
• Extension with interference
• Spread out, unreliable connections
• Sensor placement algorithms
Clustered wireless networks
Random point
processAlgorithm Connectivity
each point is covered by at least a red disc and each red disc covers at least a point
Franceschetti PhD Thesis, CIT 2003
r
R
Theorem
iffor any covering algorithm, with probability one
then for c percolation occurs1
if then some covering algorithms may avoidpercolation for any value of λ, with probability one
1
R radius blue discr radius red disc r
Covering algorithms
“Covering Algorithms, continuum percolation, and the geometry of wireless networks”
Annals of Applied Probability, 2003Booth Bruck Franceschetti Meester
PhD Thesis, CIT, 2003Franceschetti
When which classes of algorithms form an unbounded connected component, a.s., When is high?
From connectivity to network capacity
• Gilbert’s model
• Extension with interference
• Spread out, unreliable connections
• Sensor placement algorithms
• Throughput capacity
Throughput capacity without interference
How many disjoint paths there are that traverse the network?
Nodes closer than a given range are connected
Previous results, routing in high density regime
Gupta Kumar (2000)
Kulkarni Viswanath (2002)
El Gamal, Mammen, Prabhakar, Shah (2004)
High density regime routing
• Divide area into small boxes • Scale down power to allow only
transmission to adjacent boxes• Route along almost straight lines
• Adopt thermodynamic scaling• Take c large to have many crossing paths of adjacent full
boxes (by percolation)• Use these paths as the “wireless backbone” to relay traffic
crossing path
Our strategy
Throughput per node vs Range
Interference limited networkNo interference ideal network
0
c
Percolation
=log n =n
High density regime
Interference
Bottom line
Scaling power at a slow rate, order, can use straight line routing
Scaling power at a fast rate, disorder, no backbone forms
Phase transition, backbone forms, rich in crossing paths, not straight lines, carry most traffic over short hops
• Gilbert’s model
• Extension with interference
• Spread out, unreliable connections
• Sensor placement algorithms
• Throughput capacity
• Closing the loop
Let’s look at the Application level
Random losses in the feedback loop
Sinopoli Schenato Franceschetti
Poolla Sastry Jordan IEEE Trans-AC (2004)
SystemSensor
web
ControllerState
estimator
WirelessMulti-hop
• What happens to the Kalman filter when some sensor readings are lost?
• Can we bound the error covariance
The road ahead
• Towards a system theory of large scale networks• Spatial stochastic networks as a core discipline• Intellectual unification across disciplines
Phase transitions
• Phase transitions are a fundamental effect in engineering systems with randomness
• Optimal operation regions are often at the boundary of these transitions
A random walk model of wave propagation IEEE Trans.-AP
Franceschetti Bruck Schulman
Stochastic rays propagationIEEE Trans.-AP
Franceschetti
Some more work…
Interaction with physical level
Small world networksSmall-world networks a continuum model
Franceschetti Meester
A geometric theorem for network designIEEE Trans.-Comp.
Franceschetti Cook Bruck
Lower bounds on data collection times in sensory networksIEEE-JSAC
Florens Franceschetti McEliece
A group membership algorithm with a practical specificationIEEE Trans.-PDS
Franceschetti Bruck
Algorithms and protocols