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

2 Self-similar feature of traffic n Fractal characteristics u order of dimension = fractal n Self-similar feature u across wide range of time scales n Burstness: across wide range of time scales n Long-range dependence u ACF (autocorrelation function) n Power law spectral density n Hurst (self-similarity) parameter 0.5<H<1 n [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.

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

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, Y t is a FDN (fractionally differenced noise) --FARIMA(0,d,0)

Generating FARIMA Process for Model-driven Simulation

Table 1: H and d of 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

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

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 X t.  Step 2: Obtaining an approximate value of d according to the relationship d = H Three method to obtain H: - Variance-time plots - R/S analysis - Periodogram-based method

Building a FARIMA(p,d,q) Model to Describe a Trace(Cont.)  Step 3: Doing fractional differencing on X t. 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,,

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

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

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

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

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

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

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

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

Traffic Prediction  adapted h-step forecast by adding a bias where e t (h) the forecast errors, and u the upper probability limit.

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

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.

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

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