Post on 19-Apr-2018
Capacity estimates of wireless networks in
Poisson shot model over uniform or fractal
Cantor maps
Philippe Jacquet
Bell Labs
Simons Conference on Networks and
Stochastic Geometry
• Classic model:
– Wireless devices
Poisson distributed on
a map
– A density measure λ
Wireless networks:
Poisson shot models
On the infinite plan: F Baccelli, B Blaszczyszyn, "Stochastic geometry and
wireless networks," 2009
Access point network
architecture • emitters
transmit independent
flows toward an
access point z.
– Signal of emitter i
comes to z with
energy level
– Our objective: to
estimate the average
capacity of the system
,,,1 izz
z
zi
)( izs
The capacity estimation
problem • Three estimators:
– A shannon equivalent
capacity upper bound
– The flat outage
capacity as a classic
– The optimized outage
capacity used in 4G
z
zi
)(λSC
),( λKCF
)(λBC
Beyond Uniform Poisson models
• The world is fractal!
– Self similar scaled
• The revenge of the cauliflower
Plan of the talk
• Capacity definitions and physical models
• Infinite network maps and the dark night
paradox
• Outage capacities in infinite uniform maps
• The energy field Theorem and Shannon
capacity
• Fractal geometries
• Capacities in Cantor maps
• Conclusion
• Entertainment: the french army versus fractals
Capacity definitions and
physical model
Shannon capacity model
• « Non collaborative »
Shannon capacity – Each flow is noise for the
other flows
– Shannon capacity for
Gaussian noise :
z
zi
Hzper bit )(
)(1log)( 2
i
ij
j
iS
zsN
zsEC
Flat outage capacity model
• Classic wireless – Packets are all coded with
same rate R
– SNIR threshold K
– :
z
zi
Hzper bit )(
)(),(
i
ij
j
iF K
zsN
zsPREKC
Hertzper bit )1(log2 KR
Optimized outage capacity
• New generation
wireless – Transmitter i optimizes its
coding rate acording to
SNIR statistics
– :
z
zi
Hzper bit )()(
)()()(
i
i
ij
j
iiB zK
zsN
zsPzREC
)(),( ii zKzR
))(1(log)( 2 ii zKzR
Physical assumptions
• Signal attenuation and i.i.d. fading
• The Fi are i.i.d. (eg Rayleigh, exponential)
– Don’t change during packet transmission.
i
ii
zz
Fzs
)( 2for
Infinite network map and the
dark night paradox
Infinite plan network map
• Cumulated energy
• Distribution
– With
– eg Rayleigh fading
i
i zsNS )()(
)(exp),( SEw
)(exp][),( feEw N
2/2)( )1(][))(1()( dzeFEdzeEfzzs
1)1( 2/dze
z
2
)1(][ FE
Known results
• Shannon capacity
– Noiseless
• Flat outage capacity
• Optimized outage capacity
0N
2log
1)(
SC
2log
1)sin()1log(),(
K
KKCF
?)( BC
Horizon sharp drop
Horizon smooth drop
Sharp or smooth horizon drop?
The dark night paradox
• When α→2, S(λ) →∞.
– Thus capacities are 0 when α<2
2log
1
)(
)(
log2
1E
)(
)(1log)( 2
i
i
i
iS
S
zs
S
zsEC
16
Generalized Poisson shot model
• Terminals uniformly
distributed in
– >D
DR
2log)(
DC
physic
P. Jacquet, "Shannon capacity in poisson wireless network model," 2009
geometry information
)1(][)( DFEf
D
Outage capacities in uniform
maps
Case of Rayleigh fading
• Exponential
• Exercice:
– Proove that
• Hint:
xexFP )(
)(),( KSFrPKrp
)(exp)(),( KrfdxerK
xSPKrp x
KdzKzp
K
KCF )sin(),(
)1(log
),( 2
2
)sin()1()1(
Flat outage capacity in general
fading • In
• The same!
DR
DF dzzsKSPK
KC0)()(
)1(log
),(
2
i
i
dKff
i
)(exp)(
2
1
0
1 )(exp2
dKCCi
ee ii
K)sin(
Optimized outage capacity,
Rayleighy fading • Classic local optimization
→
Close formula
with
via classic numerical analysis
0)(,)1log(
rKrpK
K 0))((exp)1log(
rrKKK
dKKKKKCB )(')1log()(1
exp2log
1sin)(
0
)1log()1()(
1
KK
KK
eC
C
S
B 1
)(
)(lim
1
Optimized outage capacity,
Rayleighy fading
)(
)(
S
B
C
C
α Horizon sharp drop
Energy field theorem and
Shannon capacity
Le champ de tournesol, V. van Gogh
The energy field theorem
• Total energy received
• Differential form: the energy field theorem
– Independent of fading, noise, etc.
– Notice that capacity of node x is
i
izsS )()(
2
2 ))(
)(1log)( dx
S
xsECS
2
22 )(log))()((log)( dxSxsSECS
)(log)(
SECS
z
xExists with
proba 2dx
2
22 ]\/)([log]/)([log)( dxxISExISECS
)(log
SEx
Shannon capacity proof (i.i.d.
fading)
• Space contraction
– By arbitrary factor
– increases energy
• Equivalent to density increase
– By factor
)()( SaS
a
Da /1
)()( /1 SaaS D
))1((loglog))((log SED
SE
2log))((log)( 2
DSECS
z
zi
Shannon capacity: hackable
formula • Via Mellin transform
deEs
SE sSs 1
0
)( ][)(
1])([
0
))(exp()()(
dffCS ][ NeE
Fractal geometries
Euclid (-300) « Elements »
Uniform Poisson models are not
(always) realistic
• The world is fractal
– Fractal cities
Fractal models
• Fractal maps
– from fractal generators
29
Shannon capacity theorem
extended to fractal maps
– Eg Sierpinsky triangle
– Fractal dimension
– If extension holds, then
capacity increases on fractal
map
...58.12log
3logFd
?2log
)(F
Sd
C
Small periodic fluctuations of mean 0
)(logP
Fractal dimension in physics
• The probability of return of a random walk
on the Sierpinski triangle
– After n steps is (Rammal Toulouse
1983)
– generalize s the random walk in D-lattice
• Other dimension on fractals
– Spectral dimension
• Power laws in state density
• Used in percolation
2/
1Fd
n
Fractal (Hausdorf) dimension
• Sierpinski triangle
– Divide unit length by 2
– Structure is divided by 3
3
1
2
1
Fd
24
1
2
1
D
D
Fractal maps
Cantor maps
2log
2log2
adF
2log
4log2
adF
a a
Fractal dimension of Cantor
map
a
2
1
Fd
a
Poisson shot on Cantor map
Cantor maps
• Support measure
– Defined recursively (dimension 1)
– Poisson shot on Cantor map
• Mapped from a uniform Poisson shot
1
0 1
1
m FF aa
m μμ
Fμ
2101 . bbbbbx nn 1,,0 jb
j
jjb
aaxk1
)1()(
Cantor map
• Isometries can be used in the recursion
mm FmF Jaz
1
0
μμ
Capacities in Cantor maps
38
Shannon Capacity in Cantor
map • Access point at left corner, arbitrary fading
– Contraction by factor
– Density increases
a
a
)()( SaS
4
))((loglog))4((log 222 SEaSE
)(loglog4log
log))((log 2
2 Qa
SE periodic
)(log'4log
log))((log)( 2
2
Qa
SEd
dCS Periodic of
Mean zero
39
Periodic oscillation
– Small indeed (thanks to the no
free lunch conjecture)
– exact analysis via Fourier
transform over the hackable
formula
– Amplitude of order
4,3.0 a
310
P. Jacquet: Capacity of simple MIMO wireless networks in uniform or fractal maps,
MASCOTS 2013
Random access points
• Access point in the fractal map in position z
– Same contraction argument
),( zCS
)(log2log
)),((
R
F
S Pd
zCE
Small periodic fluctuations of mean 0
Shannon capacity on random
access points • Oscillation amplitude?
– Some bounds but not tight
– Simulations show small amplitudes
Fd
Philosophical consequence
• The fractal Poisson shot model never
converges toward the uniform Poisson
shot model
– Even when the fading variance tends to
infinity.
Economical consequence
• The actual capacity increases significantly
on fractal Poisson shot model
2log)(
F
Sd
C
DdF
The fractal dark night
• Straightforward analysis gives power law
distance distribution
– Therefore when
drrPrddz Fd
F
drrzr
F )(log1
12
μ )(log2
2 rPrdz Fd
rz
F
μ
)(S Fd
The fractal dark night
– Horizon is
– Sharp horizon drop!
DdF
2log
1)(lim
S
dC
F
F
F
d
The flat outage capacity
• We have
• With Rayleigh fading
– The quantity is periodic in K and λ
and is when
• Smooth horizon drop.
• Should hold with general fading
)(log)1(][ 3 PFE FF
F
2
2
)(exp)1(log
),(dzKrf
K
KCF
F μ
drrPrdrKPrK FFF d
F
d
F
F )()log(logsin
exp 1
1
0
3
)1log(
),(
K
KCK FF
1FO 1F
2)( ])[1()( dzeEf F
zsμ
Optimized outage capacity
• We prove (easy) that is periodic in λ
– Classic contraction argument.
– We prove (hard) that its mean value satisfies
• Sharp horizon drop, like with uniform Poisson.
)(BC
eC
C
S
B
F
1
)(lim
1
Conclusion
• Shannon capacity and optimized outage
capacity have sharp horizon drops
• Capacities on fractal maps are much
larger than capacities on uniform maps
• capacity estimates have small periodic
oscillations
• Generalization to self-similar geometries?
Thank you!
Questions?
French Army fractal tactical
organization
Xxx
corps1
Xxx
corps2
Xxx
corps3
Xxx
corps4
X
brigade Bernadotte
Davout
Lanne
Augerau
Napoléon
Fractal sets: Disposition in diamond of French Army (4 corps)
X
brigade1
X
brigade2
X
brigade3
X
brigade4
III
regiment
Disposition of French « Corps » in four brigades
II
bataillon1
II
bataillon2
II
bataillon3
II
bataillon1
II
bataillon2
II
bataillon3
Organization of French Brigades in two regiments
I
compagnie1
I
compagnie2
I
compagnie3
I
compagnie4
section
French batallion in four companies
section1
section2
section3
section4
The French company in four sections
escouade1
escouade2
escouade3
escouade4
The French section in four « escouades »
French escouade of 16 soldiers