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.

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

FARIMA Models

,)()(tt

d aBXB

21 2( ) 1 ,ppB B B B

,1)( 221

qqBBBB

,)()1(0

k

kdd BkdB

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)

,)()(1

ttYBBX

.t

d

taY

Direct definition method:

- step1: Generating a FDN Yt using (2-4).

- step2: Generating a FARIMA(p,d,q) process Xt

using (2-3). Another method, by Hosking:

- step1: Generating a FDN Yt using Hosking

algorithm.

- step2: Generating a FARIMA(p,d,q) process Xt

using (2-3).

Generating FARIMA Process for Model-driven Simulation

Table 1: H and d of generated FARIMA(0,d,0) and after fractional differencing

FARIMA(0,d,0)

Definition algorithm Hosking algorithm

after fractionaldifferencingtarget

parameterd H d H d H d

0.1 0.614 0.114 0.613 0.113 0.521 0.021

0.2 0.721 0.221 0.714 0.214 0.540 0.040

0.3 0.819 0.319 0.821 0.321 0.537 0.037

0.4 0.924 0.424 0.911 0.411 0.544 0.044

0.45 0.961 0.461 0.966 0.466 0.543 0.043

Verification Experiments

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 Xt, we can obtain from (2-1)

( 3-1 )

where

( 3-2 )• Fractional differencing

• Using the known ways for fitting ARMA models

td

t XW

tt aBBW )()(1

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

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, ,

00

)(j

jtjtk

kt

dt XXBk

dXW

)1()1(

)1()1(

jdj

dj

j

1 2, , p 1 2, , q

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

Table 1: Fitted FARIMA models of CERNET and Bellcore traces

FARIMA(p,d,q) model parameterstrace

p 1 d q

1

PAug.TL 1 -0.1714523 0.2949911 1 -0.3738780

pOct.TL 1 -0.1511854 0.2520007 1 -0.2518229

C1003 1 0.3040983 0.1978055 0

C1008 1 -0.0439937 0.1834420 0

Feasibility Study (Cont.)

Simplification Methods of Modeling

Feasibility Study (Cont.)

Fixed order (sample about 100s) Simplifying the modeling procedure

ExperimentsFARIMA(p,d,q) model parametersData set

p 1 d q

1

Aug1 1 -0.1714523 0.2949911 1 -0.373878

Aug2 1 -0.1143173 0.2783244 1 -0.305762

Aug3 1 -0.1853628 0.3515382 1 -0.329502

Aug4 1 -0.1763141 0.2813467 1 -0.306262

Aug5 1 -0.1599587 0.2995875 1 -0.280055

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

0jjtjt aX

0jjtjt Xa

d

jj BBB j

)()( 1

0

d

jj BBB j

)()( 1

0

Prediction (Cont.)Minimum mean square error forecasts (h-step)

where

The mean squared error of the h-step forecast

1

)( ˆ)(ˆj

jhthjt XhX

1 ,)(1

11

)(

hih

j

h

iihj

hj jj )1(

1

0

2222 ))(ˆ()(ˆh

jjthtt hXXEh

Prediction for Actual Traffic

Feasibility Study

the h-step forecasts , FARIMA(1,d,1) vs. AR(4)

0

1

2

3

4

5

6

7

8

9

1

Time

Fore

cast

Valu

es

Trace

FARIMA Forecasts

AR Forecasts

Mean

t-50 t+50t t+100t-100

Prediction for Actual Traffic (Cont.)

Feasibility Study (Cont.)

one-step forecasts vs. actual values, time unit = 0.1s

0

10

20

30

40

50

60

70

80

1 11 21 31 41 51 61 71 81 91

Time Unit = 0.1 Second

Packets

/Tim

e U

nit

Trace

Forecasts

68% Limits

80706050403020 10090100

Traffic Prediction

adapted h-step forecast by adding a bias

where

et(h) the forecast errors, and u the upper probability limit.

utut hXhX )(ˆ)(ˆ

10.5 ,](h)P[ uue ut

Traffic Prediction (Cont.)

Adaptive traffic prediction of traceNormal confident interval

forecast error <= t(1)when probability limit = 0.6826 (~32%)

forecast error <= 2t(1)when probability limit = 0.9545 (~4.5%)

Adapted confident interval bias u = 0 when u = 0.5bias u = t(1) when u = 0.8413 (~16%)bias u = 2t(1) when u = 0.97725 (~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

Adaptive traffic prediction of trace, time unit = 0. 1s

0

10

20

30

40

50

60

70

80

90

100

1 11 21 31 41 51 61 71 81 91

Time Unit = 0.1 Second

Packets

/Tim

e U

nit

Trace

Forecasts

84%, 98% Upper Limits

80706050403020 10090100

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