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1 BellCore Ethernet variable bit rate VBR fractal self-similarity 994 IEEE/ACM Transactions on Networking On the self-similarity nature of Ethernet traffic extended version LAN WAN CCSN/SS7 ISDN ATM VBR cluster burst 224

2 ATM chaos long-range dependence LRD 6. Mbps 4000bit 4ms

3 ms ms 5 20ms ms

4 6-6-(a) 4ms 6-(b) 6- Cantor 6-2 Cantor 0 /3 /3 227

5 6-2 Cantor 5 S i i Cantor S 0 = [0 ] S = [ /3] [2/3 ] S 2 = [0 /9] [2/9 /3] [2/3 7/9] [8/9 ] Cantor /3 Cantor 228

6 Cantor T g(t) g(t) = g(t+at) a=

7 -3(a) 6-3(b) 6-3(b) x(t) H 0.5 H a 0 a -H x(at) x(t) 3 E E = [ x() t ] [ x( at) ] H a Var [ ()] [ x( at) ] Var x t = a R ( t,s) x R ( at,as) x = a 230

8 6-3 H Hurst H persistence long-range dependence LRD H [0.5 ] H=0.5 H fractional Brownian motion FBM FBM FBM B H (t) B H (t) = Xt H t H 23

9 X 0 H E[X]=0 E[B H (t)]=0 X a Var(aX)= a 2 Var(X) Var[B H (t)]= Var[t H X]= t t B H (t) X B H (t) H=0.5 FBM FBM f B H ( x,t) = 2 x / 2t 2πt e 2 H Var[B H (t) B H (s)] = E[(B H (t) B H (s)) 2 ] = t s 2 B H (t) R ( t,s) E[ B ( t) B ( s) ] E[(B H (t) B H (s)) 2 ] = E[B 2 H(t) + B 2 H(s) 2B H (t)b H (s) ] E 2 [ B () t B ( s) ] = E B () t H H = = B H = H H 2 2 ( [ ] + E[ B () s ] E[ ( B () t B () s ) ]) H ( Var[ B () t ] + Var[ B () s ] Var[ B () t B () s ]) H H H ( t + s t s ) FBM 3 B H (at) B H (t) FBM B H (at)=x (at) H E[B H (at)] = a H E[B H (t)]=0 E[B H (t)]= E[B H (at)]/ a H Var[B H (at)]= Var[X(at) H ]= (at) Var[X]= (at) Var[B H (t)]= Var[B H (at)]/ a H H H H 232

10 R BH 2 a = 2 2 = a ( at as ) ( t + s t s ) ( at,as) = ( at) + ( as) H R BH ( t,s) FBM -t 0 0 t [( BH ( 0) BH ( t) )( BH ( t) BH ( 0) )] E[ BH ( t) BH ( t) ] ( t) + t t t ) E = = = 2 2 ( 2t) t H=0.5 H 0.5 FBM t H t t 2 233

11 X(t) {x t t=0 2 } { } x m ( m ) ( m x = x ),k = 0,, 2,L ( m) xk = m x (3) ( ) x x = km i= km xi ( m ) + x 3 3k 2 3k 3k k 3 + x x () x (3) 3 3 x (m) x m 0 x exactly self-similar m= 2 k Var(x (m) ) = Var(x) / m R ( m ) ( k) R ( k) x = x Hurst H H= 2) = /m 0 234

12 0 x asymptotically self-similar k ( ) Var ( ) ( x) Var x m = β m R ( m ) ( k ) R ( k), m x x m 0 R (m) ( ) 0 m R( ) (b) m 6-2-(a) x (m) m /m / m /m x (m) m m m m 235

13 6.2.2 long-range dependence LRD Joseph effect 0 C( ) L L C( ) = R( ) 2 = 2 2 = 0 short-rang dependent C(k) a k k 0 a k= 0 x k =, x x < k C( k) ( k) β C ~ k k 0 236

14 Hurst H= /2 k C ( k) = 2 spectral density S ( ω ) R( τ ) j ωτ d = e τ 2 f j= S( ) x(t) ( ) ( ) S 0 = R τ dτ S(0) R( ) R( ) S jkω ( ω) = R( k) e S( 0 ) = R( k) k = k= S(0) R( ) 237

15 S ( ω ) ~, ω 0 0 < γ < γ ω H 0 /f 0 3 heavy-tailed distribution X F(x) = P{X x} x a x 0 a high variability Noah k a k a Pareto f(x) = F(x) = 0 x k 238

16 f a k k x a+ k x ( x) =, F( x) =, x > k,a > 0 a E a a [ X ] = k, a > k a a 2 a 6-4 Pareto Pareto 6-4 Pareto 239

17 WWW Pareto-Train ON/OFF Taqqu {X x} x 2 renewal reward process B H (t) H= 3 /2 Pareto-Train ON/OFF ON Willinger LAN Ethernet Leland Willinger Bellcore Ethernet LAN Ethernet Hurst H=0.9 Willinger Pareto ON/OFF Ethernet 240

18 ON OFF ON OFF ON OFF Ethernet Willinger ON/OFF Pareto a 2 Willinger Pareto ON/OFF Hurst H= 3 a /2 a H Ethernet a=.2 H=0.9 2 WWW Cover 37 Web Web 50 Web Ethernet Web Web Pareto ON/OFF a.6.5 Web 3. 7 Duffy 7 SS7(Signaling System Number 7) ISDN SS7.7 SS7 SS7 24

19 4 TCP FTP TELNET Paxson TCP TCP FTP TELNET TCP TELNET FTP TELNET FTP 5 VBR ATM Internet Garret 2 JPEG / Pareto Garret 242

20 Beran 20 VBR VBR 6 Deane LAN Deane 7 TCP FTP Video step function log-normal 6.3 Ethernet/ISDN Bellcore 243

21 Ethernet ISDN % 50% 60% Ethernet Leland Willinger Ethernet Hurst H 244

22 Ethernet ATM 3 Norros Norros FBM 6.2 FBM q q ρ 2 ( H ) = H ( H ) ( ρ ) H Hurst H=0 5 q= ( ) M/M/ M/D/ ρ ρ q = ρ 2 2 ( ρ) 6-6 H=0 9 H=0 75 M/M/ M/D/

23 H

24 995 SIGMETRICS Ryu Ryu VBR TCP VBR admission control Qos TCP Web FTP 247

25 6.4 H x (m) m ( m) Var ( ) ( x) Var x ~ β m H= 2 log[var(x (m) )] log[var(x)] log(m) log[var(x)] m Var(x (m) ) m - m x(t) - 0 H 2 R/S x(t)={ x t t = 0 2 } x(t) N R/S R S max j N = j k= ( X M ( N )) min ( X M ( N )) k N N ( X k M ( N )) j= j N j k= 2 k 248

26 M(N) N M N N ( N ) = j= X j R/S N R/S N/2 H H 0 5 log[r/s] Hlog[N] Hlog2 [R/S] N - H 3 Whittle R/S H (H 0 5) (H 0 5) x(t) S( ) x(t) ( k) = E[ x( t)( x t k) ] S( ω ) = R( k) R + k e jωk 249

27 Rˆ N N N n= 0 ( k) = X ( n + k) X ( n) S( ) R(k) x(t) { x t t = 0 2 } N I N ( ω ) N = xke 2πN k= jkm periodogram H H S( H) H H ππ S ( ω ) ( ω,h ) I N d ω 2 Whittle {x k } N 2 /N 4 /N 2 H log S ( ) ( ω ) Var H ) = 4π π π H 2 dω Whittle R/S 250

28 H Whittle H 6.5 M/G/ X t t {X t t=0 2 } (asymptotically self-similar) M/G/ random midpoint displacement RMD) SPARC station

29 Garrett ARIMA n O(n) 2 Leland Taqqu TCP ON/OFF ON/OFF ON ON/OFF Paxson Floyd independent identically-distributed i.i.d Pareto pseudo-self-similar FBM A(t) = mt + αm Z(t) t ( ) A(t) t Z(t) FBM 0 5 H m a Norros P{L x} Weibull distribution 252

30 β γx P{ L > x} ~ e, 0 < β < a m H FBM Likhanov M/G/ bursts N (t) Pollaczek Khintchine formulae M/G/ Andersen Nielsen two state MMPP Markov modulated Poisson process superposition LRD 3 4 MMPP 4 5 MMPP SPP switched poisson process MMPP MMPP Markovian arrival process MAP Markov MMPP FBM 253

31 254

➀ ➁ ➂ ➃ ➄ ➅ ➆ ➇ ➈ ➉ Lecture on Stochastic Processes (by Lijun Bo) 2

Stochastic Processes stoprocess@yahoo.com.cn 111111 ➀ ➁ ➂ ➃ ➄ ➅ ➆ ➇ ➈ ➉ Lecture on Stochastic Processes (by Lijun Bo) 2 : Stochastic Processes? (Ω, F, P), I t I, X t (Ω, F, P), X = {X t, t I}, X t (ω)