Transition density estimates for diagonal systems of SDEs ...
Capacity estimates of wireless networks in Poisson … estimates of wireless networks in ......
Transcript of Capacity estimates of wireless networks in Poisson … estimates of wireless networks in ......
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