Capacity estimates of wireless networks in Poisson shot model over uniform or fractal Cantor maps. Philippe Jacquet Bell Labs

Transcription

1 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

2 Classic model: Wireless networks: Poisson shot models Wireless devices Poisson distributed on a map A density measure λ On the infinite plan: F Baccelli, B Blaszczyszyn, "Stochastic geometry and wireless networks," 2009

3 Access point network architecture z,, z, emitters 1 i transmit independent flows toward an access point z. Signal of emitter i comes to z with energy level s ( z i z i z Our objective: to estimate the average capacity of the system

4 The capacity estimation problem Three estimators: A shannon equivalent capacity upper bound C S (λ z i The flat outage capacity as a classic C F ( K, λ z The optimized outage capacity used in 4G C B (λ

5 Beyond Uniform Poisson models The world is fractal! Self similar scaled The revenge of the cauliflower

6 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

7 Capacity definitions and physical model

8 Shannon capacity model «Non collaborative» Shannon capacity z i Each flow is noise for the other flows Shannon capacity for Gaussian noise : z C S ( Elog 2 i 1 N s( z ji i s( z j bit per Hz

9 Flat outage capacity model Classic wireless Packets are all coded with same rate R SNIR threshold K : z z i bit per Hz ( (, ( i i j j i F K z s N z s P R E K C bit per Hertz (1 log 2 K R

10 Optimized outage capacity New generation wireless Transmitter i optimizes its coding rate acording to SNIR statistics : z z i bit per Hz ( ( ( ( ( i i i j j i i B z K z s N z s P z R E C (, ( i i z K z R ( (1 log ( 2 i i z K z R

11 Physical assumptions Signal attenuation and i.i.d. fading s i ( zi for 2 z z The Fi are i.i.d. (eg Rayleigh, exponential Don t change during packet transmission. F i

12 Infinite network map and the dark night paradox

13 Infinite plan network map Cumulated energy S ( N s i ( z i Distribution w(, E exp S( With N w(, E[ e ]exp f ( f eg Rayleigh fading s ( z 2 / z 2 ( (1 E( e dz E[ F ] (1 e dz (1 e / z dz 2 1 E[ F ] (1 2

14 Known results Shannon capacity Noiseless C S ( 1 log 2 N 0 Flat outage capacity C F (, K log(1 K K sin( 1 log 2 Horizon sharp drop Horizon smooth drop Optimized outage capacity C B (? Sharp or smooth horizon drop?

15 The dark night paradox When α 2, S(λ. Thus capacities are 0 when α<2 C S s( z 1 ( ( log 21 i s z E ( i E log2 ( i S i S 1 log 2

16 Generalized Poisson shot model Terminals uniformly D distributed in R f >D D ( E[ F ] D (1 physic C( D log 2 geometry information P. Jacquet, "Shannon capacity in poisson wireless network model,"

17 Outage capacities in uniform maps

18 Case of Rayleigh fading Exponential P( F x e x Exercice: Proove that Hint: p( r, K P Fr KS( p( r, K PS( x K r e x dx exp f ( Kr C F (, K 2 sin( p( z, K dz K log (1 K 2 (1 (1 sin(

19 Flat outage capacity in general fading In The same! D R F dz D z s KS P K K C 0 ( ( (1 log, ( 2 i i d K f f i ( exp ( ( exp 2 d K C C i e e i i K sin(

20 K Optimized outage capacity, Rayleighy fading Classic local optimization K pr, K( r 0 log( 1 exp ( ( K K r r 0 log( 1 C B Close formula ( with sin 1 log 2 ( K 0 K 1 exp K 1 K (1 Klog(1 K ( K log(1 via classic numerical analysis lim 1 C C B( 1 S ( e K '( K dk

21 Optimized outage capacity, Rayleighy fading C C B S ( ( Horizon sharp drop α

22 Energy field theorem and Shannon capacity Le champ de tournesol, V. van Gogh

23 The energy field theorem Total energy received C S C S C S Differential form: the energy field theorem Independent of fading, noise, etc. Notice that capacity of node x is S ( s( z i i s( x 2 ( E log 2 1 dx S( ( E log ( S( s( x log S( dx 2 2 x C S 2 z x ( E[log S( / I ] E[log S( / I \ dx 2 2 ] x Exists with proba ( Elog S( Elog S( x 2 dx 2

24 Shannon capacity proof (i.i.d. fading Space contraction By arbitrary factor increases energy Equivalent to density increase By factor a S( a S( S( a D a 1/ 1/ D a E(log S( ( log E(log S(1 D C S ( E(log 2 S z i z S( Dlog 2

25 Shannon capacity: hackable formula Via Mellin transform d e E s S E s S s 1 0 ( ] [ ( 1 ] ( [ 0 ( exp( ( ( d f f C S ] [ N e E

26 Fractal geometries Euclid (-300 «Elements»

27 Uniform Poisson models are not (always realistic The world is fractal Fractal cities

28 Fractal models Fractal maps from fractal generators

29 Shannon capacity theorem extended to fractal maps Eg Sierpinsky triangle Fractal dimension d F log 3 log If extension holds, then capacity increases on fractal map C S ( d F? P(log log 2 Small periodic fluctuations of mean 0 29

30 Fractal dimension in physics The probability of return of a random walk on the Sierpinski triangle 1 After n steps is (Rammal Toulouse / n d F generalize s the random walk in D-lattice Other dimension on fractals Spectral dimension Power laws in state density Used in percolation

31 Fractal (Hausdorf dimension Sierpinski triangle Divide unit length by 2 Structure is divided by d F D 1 4 D 2

32 Fractal maps Cantor maps a log 4 log 2 d 2 d F 2 2 F 2 log a log a a

33 Fractal dimension of Cantor map a d F 1 2 a

34 Poisson shot on Cantor map

35 Cantor maps Support measure Defined recursively (dimension 1 μ F 1 a 1 m m0 1 μ F μ a Poisson shot on Cantor map Mapped from a uniform Poisson shot x b nbn 1 b. b b2 0,, F b j k( x (1 a j a j b j 1

36 Cantor map Isometries can be used in the recursion μ F 1 m0 z m μ F aj m

37 Capacities in Cantor maps

38 Shannon Capacity in Cantor map Access point at left corner, arbitrary fading a Contraction by factor S( a S( Density increases 4 E(log S(4 log 2 a E(log 2 S( 2 log 2 a E(log 2 S( log Q(log log 4 d log 2 a ( E(log S( Q'(log d log 4 C S a periodic 2 Periodic of Mean zero 38

39 Periodic oscillation Small indeed (thanks to the no free lunch conjecture exact analysis via Fourier transform over the hackable formula Amplitude of order 10 3 a 0.3, P. Jacquet: Capacity of simple MIMO wireless networks in uniform or fractal maps, MASCOTS

40 Random access points Access point in the fractal map in position z (, z C S Same contraction argument E( CS (, z PR (log d log 2 F Small periodic fluctuations of mean 0

41 Shannon capacity on random access points Oscillation amplitude? Some bounds but not tight Simulations show small amplitudes d F

42 Philosophical consequence The fractal Poisson shot model never converges toward the uniform Poisson shot model Even when the fading variance tends to infinity.

43 Economical consequence The actual capacity increases significantly on fractal Poisson shot model C S ( d F log 2 d F D

44 Therefore S( when df The fractal dark night Straightforward analysis gives power law distance distribution r z μ rdr dz d r P1(log r 2 df 1 F F dr z μ r F dz 2 r df P 2(log r

45 The fractal dark night Horizon is lim d F C S ( 1 log 2 d F D F df Sharp horizon drop!

46 C log F We have The flat outage capacity With Rayleigh fading 2 f F F The quantity log(1 K is periodic in K and λ and is O 1 when 1 Smooth horizon drop. Should hold with general fading F F E[ F ] (1 P3 (log (, K 2 exp f ( Kr μfdz (1 K F F d F df 1 exp K r P3 (log K log r dfr P1 ( r dr sin 0 F C (, K K F s ( z 2 ( (1 E[ e ] μfdz F F

47 Optimized outage capacity We prove (easy that C B ( Classic contraction argument. is periodic in λ We prove (hard that its mean value satisfies CB lim F 1 C ( S 1 e Sharp horizon drop, like with uniform Poisson.

48 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?

49 Thank you! Questions?

50 French Army fractal tactical organization

51 Fractal sets: Disposition in diamond of French Army (4 corps Xxx corps2 Davout Xxx corps1 Bernadotte X brigade Xxx corps3 Lanne Napoléon Xxx corps4 Augerau

52 Disposition of French «Corps» in four brigades III regiment X brigade2 X brigade3 X brigade4 X brigade1

53 Organization of French Brigades in two regiments II bataillon1 II bataillon1 II bataillon2 II bataillon2 II bataillon3 II bataillon3

54 French batallion in four companies I compagnie2 I compagnie3 section I compagnie1 I compagnie4

55 The French company in four sections section2 section3 section1 section4

56 The French section in four «escouades» escouade2 escouade3 escouade1 escouade4

57 French escouade of 16 soldiers