Approccio Statistico all'analisi di Sistemi Caotici e Applicazioni all'ingegneria dell'informazione

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1 Arocco tatstco all'anals d stem Caotc e Alcazon all'ingegnera dell'informazone Ganluca ett 3 Rccardo Rovatt 3 D. d Ingegnera Unverstà d Ferrara D. d Elettronca, Informatca e stemstca - Unverstà d Bologna 3 Centro d Rcerca su stem Elettronc er l Ingegnera dell Informazone e delle Telecomuncazon Ercole De Castro (ARCE) Unverstà d Bologna cuola d Dottorato Ingegnera dell Informazone -5 ettembre 3 Unversty of Bologna Unversty of Ferrara

2 How far wll we clmb? reroducton of networ traffc self-smlar rocesses generalzed tensor aroach countable PWAM

3 = a > a > a > M [ ] [ ] :,, lm a = x a+ a + ( a a) f x ] a+, a] a a+ M( x) = x a f x ] a, a] a a A Ma, a roblem Good news: X [, ] = s a Marov artton a a Bad news: the roecton of the Perron-Frobenus oerator onto the sace of functons does not yeld a fnte-dmensonal oerator 3

4 A roblem, a (artal) soluton = [,] ' = '' = = M ( ) M = = = certan certan X X x ', x'' ϕ( x') = ϕ( x'') 4

5 The Proect-and-Factor method = < < < m [ ],, m P ρ ξ sequence of observaton nstants ont densty of the vector when ξ s dstrbuted accordng to M m ( ) ξ M ξ ξ ρ roecton Pr = H =,,,,,,, m m m m m m P [ ] ρ ξ dξ,,, m corresondng vsted macro-states 5

6 The Proect-and-Factor method + = + + a transton between two macro-states Under sutable assumtons ξ{ } =, ( ξx ξ xy x+ ξy) factorzaton m P,, m [ ] ρ ξ dξ = ρ ξ ξ ρ ξ ξ { } µ ( ) [ ] { } { } [ ] { },,, +,, m P d P d,,, m, m Pr + m m m 6 Pr + m Pr m +

7 Gven the observables only transtons between macro-states matters 7 + = + =,, s Pr +,, P m = m m [ ] ρ ξ dξ = d, {, } [ ] { },, P ρ ξ ξ Pr s,,, P s Pr = µ = , s +,, s m s P µ ( ) Pr m s s+ s s + s m s s s+ m s The Proect-and-Factor method [ ] { m} d, {, } s [ ] { } ρ ξ ξ L, d, {, m} ρ ξ ξ s = s + + ( ) ( ) + + M, s m ( ) m s

8 The Proect-and-Factor method 4 H,,,,, m = m ( ) s L, M m = + +, s m s factors are tme-deendent matrces Probablty of beng n u to t= and then leave at t=, stayng there, L, M Probablty of stayng n for tme stes and then leave, gven that we were n mmedately before Probablty of stayng n for at least tme stes, gven that we were n mmedately before L, ( ) = M, ( ) = 8

9 M L ( l) =,, l= = µ ( ) L () l l= + = µ + ( ) = µ ( ) ( ) = ( ) M ( ) ( ) M L s enough for a comlete characterzaton Proertes of the factors t = t = () L l L M H 9

10 Characterstc quanttes Macro-state asymtotc nvarant robabltes µ ( ) = = Observable average [ x ] = E, ϕ ϕ = Average macro-state soourn tme T ( ) µ = µ ( )

11 H ', '' = l= l l > l + l = ' () l I ' '' ( l ) ( l ) + M l l, l3> l + l + l = 3 ' ' '' ( l ) ( l ) ( l ) + L M l, l, l3, l4> l + l + l + l = 3 4 ' ' '' 3 s = t = t = ' = '' ( l ) ( l ) ( l ) ( l ) + L L M + ' ' 3 '' 4 t = t = l ' What f we do not now s? s = l '' s = t = l t = l l 3 ' '' s = 3 t = l t = l l 3 l 4 ' ''

12 ( l) = l= A B = A l B l = A l B l H = ', '' l= l l> l + l = ' '' ' '' ' '' l l l ' () l I ' '' ( l ) ( l ) + M l l, l3> l + l + l = 3 ' ' '' = + = ( l ) ( l ) ( l ) + L M l, l, l3, l4> l + l + l + l = 3 4 A = I A = I A ' ' '' 3 ( l ) ( l ) ( l ) ( l ) + L L M ' ' 3 '' 4 H = Whac the ndexes! I + I M + I L M + I L L M

13 A very comact form H = I + I M + I L M I L L M H = I + I L M ( s ) s= Z-transform of a generc matrx deendng on tme + + [ A] A A Z z = z = z = z [ A B] = [ A] [ B] Z z Z z Z z H = I + I L M s ( z) ( z) ( z) ( z) ( z) s= 3 ( z) = ( z) + ( z) ( z) ( z) H I I I L M

14 M ( ) = + M L( l) l= = Proertes of the factors n the z-doman ( ) = ( ) M L s enough for a comlete characterzaton L M ( ) H ( z) L L M( z) = z M = L () () ( z) z( ) = L ( z) L( ) z 4

15 Hgher-order ont robabltes H,, m Any number of macro-state transtons must be consdered between any two observaton nstants s =,,, =,,, m 5 uch a combnatoral exloson cannot be tamed but only coed wth by means of a systematc constructon rocedure

16 A monster-mang rece am and buldng blocs 6 a A (,, ) = ( ) ( z ) + z a z z z (,, ) = ( ) m m = m τ = τ = = H z, z,, z,, z τ H( τ τ ) = = ( z z) = + A ( z ) z z z ( z) = ( z) J I L = = ( z z) = m a can be ether or A can be ether L or M

17 A monster-mang rece let t wor! 7

18 An examle wth two macro-states L ( τ ) ( τ ) = ( τ ) ( τ ) = robablty of stayng n exactly τ tme stes a a = l= + () l = l= + () l 8

19 L ( z) = ( z) ( z) ( z) ( τ ) = τ = z τ + + z ( z) ( z ) U ( z ) = T z + U z α The factors α α ], ] J ( z) = ( z) ( z) ( z) ( z) M ( z) = z ( z) ( z) z z = z z ( z)( T + T ) ( ) 9 z z = z + T z z + T z ( z) ( T + T ) ( ( ) ( ))

20 The observables E ( ),, [ ϕ x ] = ϕ = ϕ () = lm z = lm z z = T T z ( ) ( ) z T + T z T+ T [ ] E ϕ( x ) = Tϕ + Tϕ = ϕ = ω T ϕ = ω T T + T T + T

21 Exonentally dstrbuted soourn tmes τ ( τ) ( π ) π = a a a = l= + () l = l= + () l a a + a + a + a a a + = = a a + π + π π π π

22 Exonentally dstrbuted soourn tmes τ ( τ) ( π ) π = π = π z ( z) C [ x x ] τ = E ϕ ϕ = ϕ ϕ, H τ τ, H ( z) C z = ϕ ϕ = ω ( π )( π ) ( π + π ) z π + π z C ( π )( π ) ( ) ( π + π ) = + τ ω π π τ π π π + π > + π = + π < low-ass whte hgh-ass

23 nd-order elf-smlarty y () = y y y ( A) A = A l= y A+ l ( 5) y ( A ) ( A) ( A) C τ E y y τ = 3 ( A C ) ( τ ) C C C C ( xa ) ( ) ( A ) ( ) ( A ) ( τ ) ( A ) ( ) τ τ β x β ],[ x > β τ A τ f the Central Lmt Theorem holds for y C C ( xa ) ( ) ( A ) ( ) () ( σ ) xa () ( σ ) A C C ( xa ) ( ) ( A ) ( ) x

24 Polynomally decayng correlatons A A ( A ) ( A) ( A) C ( τ ) = E y yτ = E y yaτ+ A = = A A C C A ( A ) ( ) = ( ) = = A C = + A A = ( m ) C A C + ( m ) C d A A ( τ) C = Kτ β β ( ) K ( β) + ( β) K + A A ( β)( β) ( β)( β) C A A A β 4 C C ( xa ) ( ) ( A ) ( ) x β

25 5 A A Polynomally decayng correlatons τ = τ + τ ( A ) C C A A, A = = A A = = A A = + A ( Aτ ) A A A / A/ A ( A β βc ) ( τ ) β A ( τ + ξ η) dξdη τ A K A β β Aτ + dd ξ = η = ( + ) β β τ ξ η dξdη ( ) β ( τ + ξ η) β ( τ + η) ( τ η) = A dξ = A β ( β)( β) ( β)( β) β η= β η= β η= { β } β β τ + τ + ( τ ) C η= η= η= β β β A A τ = β β β β

26 Polynomal decay and z-transform Tauberan roertes C ( z) = C( τ ) z τ τ = Rate of decay of C(τ) behavour of ts z-transform for z C ( τ) τ β ff C ( z) features an algebrac sngularty n z= wth W ( z ) β β ],[ C z A.M.Odlyzo, asymtotc enumeraton methods n Handboo of Combnatorcs MIT Press 6 H.. Wlf, Generatngfunctonology, Academc Press

27 Polynomal decay and soourn tmes bac to two-states systems C z = ω ( π )( π ) ( π + π ) z π + π z ngularty only n the degenerate case but t would be a smle ole! π + π = π = π = ystems wth exonentally dstrbuted soourn tmes are not self-smlar 7

28 ( z) T ( z ) U ( z ) α + α ], ] Polynomal decay and soourn tmes bac to two-states systems For exonentally dstrbuted soourn tme General exanson n a neghborhood of z= U α = = () = z ( π ) Ths s what we need ω α α 3 ( ) + ( ) C z T U z T U z ( T + T ) W z ( ) { α α } mn, { α α } mn, ],[ 8

29 Polynomal decay and soourn tmes bac to two-states systems 3 ( z) T ( z ) U ( z ) α + Hence, ( z) U α ( α )( z) z α α ], [ ( z) z dverges z τ ( z) z τ ( τ ) ( τ) z τ ( τ) τ= τ= must dverge α τ τ α ], [ 9

30 elf-smlarty/polynomal correlaton decay/oourn tmes α ( τ) τ α ], [ α β β ( ) = τ( ) τ C z W z W z α = β + β ],[ C ( τ) τ β 3 C C C C ( xa ) ( ) ( A ) ( ) ( A ) ( τ ) ( A ) ( ) β x β ],[ x > τ β A τ

31 A thrd alcaton: modelng/synthess of LAN traffc sgnals wth tunable statstcal features Chaos-based Performance modelng/synthess Otmzaton of networ traffc ON erformance ndex n terms of statstcal features of sgnals OFF 3

32 Networ traffc s self-smlar At any gven tme ste (slot) a acet may be resent (ON=) or not (OFF=). The ON-OFF rocess s nd-order self-smlar and s characterzed by the Hurst arameters H β = H ] /,[ ON and OFF states may have dfferent Hurs arameters The average of the rocess s called actvty or P ON W.E. Leland, D.V. Wlson, Hgh tme-resoluton measurement and analyss of LAN traffc: mlcaton for LAN nterconnectons, INFOCOM 99 3 K.Par, W. Wllnger, elf-smlar Networ Traffc and Performance Evaluaton, Wley,

33 ettng both H and P ON self-smlarty τ τ 4 H actvty =ON =OFF P ON T = = T = + T + = = P + P = ON ON = = 33

34 a b = = A B 4 H 4 H x y = = ( ) ettng both H and P ON = = ON x y mn + = a b s.t. P x + P y = s.t. x = s.t. y = s.t. x, y ON = = 34

35 An teratve and fnte soluton = = ON x y mn + = a b s.t. P x + P y = s.t. x = s.t. y = ON = = λ ( P ) + λ x ' = a + a λpon + λ3 y' = b + a ON s.t. x, y x' < x = y' < y = a non-trval Lemma 35

36 Have a loo at the solutons 36