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