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FARIMA( p,d,q ) Model and Application

FARIMA( p,d,q ) Model and Application. F ARIMA Models -- fractional autoregressive integrated moving average Generating FARIMA Processes Traffic Modeling Using FARIMA Models Traffic Prediction Using FARIMA Models Prediction-based Admission Control Prediction-based Bandwidth Allocation.

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FARIMA( p,d,q ) Model and Application

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  1. FARIMA(p,d,q) Model and Application • FARIMA Models -- fractional autoregressive integrated moving average • Generating FARIMA Processes • Traffic Modeling Using FARIMA Models • Traffic Prediction Using FARIMA Models • Prediction-based Admission Control • Prediction-based Bandwidth Allocation

  2. Self-similar feature of traffic • Fractal characteristics • order of dimension = fractal • Self-similar feature • across wide range of time scales • Burstness: across wide range of time scales • Long-range dependence • ACF (autocorrelation function) • Power law spectral density • Hurst (self-similarity) parameter 0.5<H<1 • [LTWW94] Will E. Leland, Murad S. Taqqu, Walter Willinger, and Daniel V. Wilson, “On the Self-Similar Nature of Ethernet Traffic (Extended Version),” IEEE/ACM Transactions on Networking. Vol 2, No 1, February 1994.

  3. FARIMA Models • A FARIMA(p,d,q) process {Xt: t =...,-1, 0, 1,...} is defined to be • (2-1) • where {at} is a white noise and d(-0.5, 0.5), • (2-2) • B -- backward-shift operator, BXt = Xt-1

  4. FARIMA Models (Cont.) • For d  (0, 0.5), p 0 and q 0, a FARIMA(p,d,q) process can be regarded as an ARMA(p,q) process driven by FDN. From (2-1), we obtain • (2-3) • where • (2-4) • Here, Yt is a FDN (fractionally differenced noise) • --FARIMA(0,d,0)

  5. Generating FARIMA Process for Model-driven Simulation

  6. Table 1: Hand dof generated FARIMA(0,d,0) and after fractional differencing

  7. Network delay on FARIMA models

  8. Network delay on FARIMA models with non-Gaussian distribution

  9. Building a FARIMA(p,d,q) Model to Describe a Trace • For a given time series Xt, we can obtain from (2-1) • ( 3-1 ) • where • ( 3-2 ) • Fractional differencing • Using the known ways for fitting ARMA models

  10. Building a FARIMA(p,d,q) Model to Describe a Trace(Cont.) • Steps of Fitting Traffic: • Step 1: Pre-processing the measured traffic trace to get a zero-mean time series Xt . • Step 2: Obtaining an approximate value of d according to the relationship d = H - 0.5. Three method to obtain H: - Variance-time plots - R/S analysis - Periodogram-based method

  11. Building a FARIMA(p,d,q) Model to Describe a Trace(Cont.) • Step 3: Doing fractional differencing on Xt . • From (2-4) we can get the precise expression • (3-3) • where • (3-4) • Step 4: Model identification: Determining p and q using known ways for fitting ARMA models. • Step 5: Model estimation: Estimating parameters (1+ p + q): d, ,

  12. Feasibility Study Constructing FARIMA Models for Actual Traffic: Traces C1003 and C1008 from CERNET (The Chinese Education and Research Network) Traces pAug.TL and pOct.TL from Bellcore Lab

  13. Feasibility Study (Cont.) Table 1: Fitted FARIMA models of CERNET and Bellcore traces

  14. Feasibility Study (Cont.) Simplification Methods of Modeling • Fixed order (sample about 100s) • Simplifying the modeling procedure • Experiments

  15. Conclusions of Building FARIMA Model • Building a FARIMA model to the actual traffic trace • Reduce the time of traffic modeling, techniques included • - fractal de-filter (fractional differencing) • - a combination of rough estimation and accurate estimation • - backward-prediction

  16. Prediction • Using FARIMA Models to Forecast Time Series • -- optimal forecasting • Assumptions of causality and invertibility allow us to write • , • where

  17. Prediction (Cont.) • Minimum mean square error forecasts (h-step) • where • The mean squared error of the h-step forecast

  18. Feasibility Study Prediction for Actual Traffic the h-step forecasts , FARIMA(1,d,1) vs. AR(4)

  19. Feasibility Study (Cont.) Prediction for Actual Traffic (Cont.) one-step forecasts vs. actual values, time unit = 0.1s

  20. Traffic Prediction • adapted h-step forecast by adding a bias • where • et(h) the forecast errors, and u the upper probability limit.

  21. Traffic Prediction (Cont.) Adaptive traffic prediction of trace • Normal confident interval forecast error <= t(1) • when probability limit = 0.6826 (~32%) • forecast error <= 2t(1) • when probability limit = 0.9545 (~4.5%) • Adapted confident interval bias u = 0 when u = 0.5 • bias u = t(1) when u = 0.8413 (~16%) • bias u = 2t(1) when u = 0.97725 (~2%)

  22. Traffic Prediction Procedure • Step1: Building a FARIMA(p,d,q) model to describe the traffic. • Step2: Doing minimum mean square error forecasts. • Step3: Determining the value of upper probability limit u according to the QoS necessary in the particular network. • Step4: Doing traffic predictions by the adapted prediction method with upper probability limit.

  23. Prediction for Actual Traffic Example Adaptive traffic prediction of trace, time unit = 0. 1s

  24. Conclusions • FARIMA(p,d,q) models are more superior than other models in capturing the properties of real traffic • Less parameters required • Possible to simplify the fitting procedure and reduce the modeling time • Good result of adapted traffic prediction for real traffic

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