Traffic Modeling and Prediction using ARIMA/GARCH model

Transcription

1 Traffic Modelig ad Predictio usig ARIMA/GARCH model Preseted by Zhili Su, Bo Zhou, Uiversity of Surrey, UK COST 285 Symposium 8-9 September 2005 Muich, Germay

2 Outlie Motivatio ARIMA/GARCH model Parameter estimatio Predictio scheme Experimets ad aalysis Coclusio

3 Motivatio (1/2) Buildig Effective Network Cotrol schemes: Cogestio cotrol, Admissio cotrol, Dyamic badwidth allocatio, adaptive rate cotrol etc. Avoidig over-, uder- estimatio. Capturig SRD/LRD characteristics. Applyig for differet time scales.

4 Motivatio (2/2) Network traffic predictio models Liear time series (AR, ARMA, etc) Oly capture SRD No-liear time series (FARIMA,TAR) Capture SRD ad LRD except multifractal Multifractal wavelet model (MWM) Not well to capture burstiess No-model based predictor (MMSE) Oly suitable for short-term predictio

5 ARIMA(r,d,m)/GARCH(p,q) model Usig ARIMA explai dyamic time series where r m d d t φ i t i t θj t j i= 1 j= 1 ( X ) = ( X ) + Z + Z Z t WN(0, σ 2 t Usig GARCH explai dyamic variace p q t = 0 + i t i + jzt j i= 1 j= 1 σ α γ σ α )

6 Parameter estimatio (1/2) ARIMA (r, d, m)/ GARCH (p, q) has 5 parameters Differeced parameter d decide statioary of time series, ca be obtaied by ACF. Order of r ad m ca be obtaied by ACF ad PACF, values ca be computed by MLE. Order of p ad q ca be obtaied by ACF ad PACF, experimet orders are set to (1,1), their clause ca be computed by MLE as well.

7 Parameter estimatio (2/2) TRAFFIC Differeced Operatio Statioary No Statioary T raffic Yes ACF&PACF MLE ARIMA Parameters Estimatio ACF&PACF MLE GARCH Parameters Estimatio Model vs T raffic accurate iaccurate END

8 Predictio scheme (1/2) Oe-step-ahead predictio predict value for ext oe time iterval K-step-ahead predictio predict values for ext k time iterval Trade-off questio predictio iterval vs. predictio accuracy predictio iterval vs. predictio legth time scale vs. predictio accuracy

9 Predictio scheme (2/2) Traffic Traiig Part Testig Part ARIMA/GARCH Modellig Model Predictor Predictio Errors SER k-step-ahead Predictio Evaluatio

10 Experimets ad Aalysis LBL-TCP-3 to be used This trace cotais two hours' worth of all widearea TCP traffic betwee the Lawrece Berkeley Laboratory ad the rest of the world. Usig ARIMA(1,1,1)/GARCH(1,1) fittig the trace All parameters of r, m, p, q are estimated by MLE

11 Model validatio: Variace-time plot (1/2) Exploratory method for LRD i large-time-scale Suppose X t is the time series with LRD, the black average over a block of legth m, the variace Var m is defied as β Var m ( cost) m L( m) The logvar m ( cost) β log m

12 Model validatio: Variace-time plot (2/2)

13 Model validatio: Multifractal spectra (1/3) Exploratory method for multifractal i small-time-scale Suppose Y is a icreasig process at time t α() t = lim α k With α Defie k k k 2 t 1 = log2 k [ Y ] [ Y ] = X (( k + 1)2 ) X ( k 2 ), k = 0,1,...,2 1 1 f ( ): limlim log () 2 2 G α = Pt α + ε α t α ε ε 0 If α>1, Xt has small icremet If α<1, Xt has istat growth called burstiess

14 Model validatio: Multifractal spectra (2/3) Partitio fuctio for multifractal spectrum = T ( q) : lim log2 E ( k [ X ]) k = 0 If q>0, burstiess If q<0, cotaiig small but o-zero icremets The multifractal spectrum ad its partitio fuctio are closely related by Legedre Trasform ( ) ( ) ( ) f ( α) = qα T q, f ' α = q, T' q = α G G q

15 Model validatio: Multifractal spectra (3/3)

16 Traffic predictio approaches Oe-step-ahead predictio K-step-ahead predictio Predictio error cotrol: predicted sigal to error rate (SER) SER 2 E[ X t ] = 10log 10 db E X t X [( ˆ 2 t ) ]

17 Oe-step-ahead result (1/3, ts=100ms)

18 Oe-step-ahead result (2/3, ts=1s)

19 Oe-step-ahead result (3/3, ts=10s)

20 SER measure of oe-step-ahead Predictio Model 100ms 1s 10s FARIMA (db) ARIMA/GARCH (db)

21 K-step-ahead result (1/3, K=1)

22 K-step-ahead result (2/3, K=10)

23 K-step-ahead result (3/3, K=100)

24 SER measure of K-step-ahead Predictio Model K=1 K=10 K=100 FARIMA (db) ARIMA/GARCH (db)

25 Coclusio ARIMA/GARCH is a good traffic model for describig SRD, LRD ad multifractal. Oe-step-ahead predictio approach is a proper predictio approach ARIMA/GARCH is a good forecastig model for etwork traffic predictio

26 Q&A Thak you for your attetio!