The Application of Fractal Process to Network Traffic Modeling

Chen Chu

South China University of Technology

1. Self-Similar process and Multi-fractal processThere are 3 different definitions for self-similar process and 2 different

definitions of multi-fractal process. Definition 1:

A continuous-time process Y(t) is self-similar if it satisfies:

Y(t) = a-HY(at) for any a>0, 0≤H<1

The equality means finite-dimensional distributions. This process can not be stationary, but it is typically assumed to have stationary increments. Fractional Brownian Motion(FBM) is such a process. The stationary increment process of FBM is FGN.

Definition I of multi-fractal:

A multi-fractal process Y(t) satisfies:

Y(t) = a-H(t)Y(at) for any a>0, 0≤H(t)<1

Multi-fractional Brownian Motion is such a process. It’s neither a stationary process nor a stationary increment process.

1. Self-Similar process and Multi-fractal processDefinition 2:

A wide-sense stationary sequence X(i). For each m = 1, 2, 3,..., Let

Xk(m) ＝ 1/m(Xkm-m+1 ＋ …＋ Xkm) ， k ＝ 1, 2, 3, …

The process X is called self-similar process if it satisfies the following conditions:

1) The autocorrelation function r(k) is a slowly varying function.

2) r (m) (k) = r(k)

If the condition 2) is satisfied for all m, then X is called exactly self-similar. If the condition 2) is satisfied only for m becomes infinite, then X is asymptotically self-similar.

FGN is exactly self-similar process.

FARIMA is asymptotically self-similar process.

1. Self-Similar process and Multi-fractal processDefinition 3 of Self-Similar process:

For a time series X and its aggregated process X(m), Let

μ(m) (q) = E | X(m) |q

If X is self-similar, then μ(m) (q) is proportional to m , so we have the following formulas:

1): log μ(m) (q) = β(q) log m + C(q)

2): β(q) = q(H-1)

Definition 2 of multi-fractal process:

β(q) is not linear with respect to q. In other words, for different q we get different H.

1. Self-Similar process and Multi-fractal processSelf-similar process

Characteristics

Definition 1 FBM: self-similar, normal distribution, stationary increments

Definition 2 FGN: self-similar, normal distribution, stationary, long-range dependent, Slowly decaying variances, Hurst effect

FARIMA: self-similar, stationary, long-range dependent, slowly decaying variances, Hurst effect, any distribution

Definition 3 FGN:

FARIMA: normal distribution.For non-normal distributional FARIMA, we get different H with different q. It is Multi-fractal process according to definition II.

1. Self-Similar process and Multi-fractal processMulti-fractal process

Characteristics

Definition I MFBM: non-stationary, non-stationary increments

Definition II (non-normal FARIMA): self-similar, stationary, long-range dependent, slowly decaying variances, Hurst effect, non-normal distribution

2. The Characteristics of Network Traffic1): Self-similarity or scaling phenomena.

However, the self-similarity exists in different scales for different network.

BC-89Aug Frame Traffic:

The scaling phenomena exists from 10ms to 100s.

2. The Characteristics of Network TrafficMAWI IPv6 WIDE

backbone. The scaling phenomena exist

during 100ms~100s. For time scale less than 100ms, there is no self-similar exist.

2. The Characteristics of Network Traffic2): The marginal distribution of network traffic is not

normal. But as the time scale increase, it becomes normal.

2. The Characteristics of Network Traffic3): The long-dependence of the traffic.

2. The Characteristics of Network TrafficBC traffic Aug89 MAWI IPv6 traffic

environment LAN, IPv4 WAN, Backbone, IPv6

Self-similarity Form 10ms to 100s Form 100ms to 100s

There is no self-similarity when time scale less than 100ms.

Distribution Almost normal Asymptotically normal

Long-dependence

The traffic show not perfect long-dependence.

When time unit is 10ms, the long-dependence is not clear.

When time unit is 100ms, the traffic show perfect long-dependence.

2. The Characteristics of Network TrafficConclusion:Larger scales (time scale larger than 10ms or

even 100ms)self-similarity long-dependencenearly normal distribution.

Small scales (time scale less than 100ms)any distribution (usually lognormal or

heavy-tail) not self-similar short-dependent

2. The Characteristics of Network TrafficModels for network traffic

For large scales: FGN is a suitable model.

For small scales: 1) generate non-normal FARIMA time

series with length n; 2) divide the FARIMA time series into k

different series, permute each of these series.

3. The Estimation of Hurst ParameterThe estimation of the Hurst parameter has a close

relationship with the marginal distribution of the time series.

Estimation of H with 100 independent FGN(10^5 long)Method FGN(H=0.5) Mean Std

FGN(H=0.9) Mean Std

R/S 0.515 0.015 0.868 0.021

Absolute moment 0.500 0.010 0.868 0.015

ACF 0.500 0.003 0.888 0.005

Aggregate variance 0.500 0.009 0.868 0.014

Different variance 0.501 0.012 0.898 0.010

Periodogram 0.501 0.006 0.905 0.006

Box Periodogram 0.497 0.011 0.856 0.011

Higuchi’s method 0.501 0.009 0.894 0.030

Peng’s method 0.501 0.010 0.900 0.013

Wavelet 0.503 0.003 0.917 0.004

Whittle 0.500 0.002 0.900 0.002

3. The Estimation of Hurst ParameterEstimation of H with 50 independent FARIMA(10^5 long)(marginal distribution is heavy-tailed with tail parameter alpha = 1.8)

Method FARIMA(H=0.6) Mean Std

FARIMA(H=0.9) Mean Std

R/S 0.609 0.012 0.871 0.018

Absolute moment 0.653 0.003 0.907 0.025

ACF 0.600 0.018 0.888 0.005

Aggregate variance 0.598 0.010 0.870 0.015

Different variance 0.605 0.034 0.881 0.061

Periodogram 0.601 0.006 0.897 0.006

Box Periodogram 0.585 0.012 0.854 0.013

Higuchi’s method 0.653 0.017 0.920 0.027

Peng’s method 0.598 0.014 0.893 0.020

Wavelet - - - -

3. The Estimation of Hurst ParameterEstimation of H with 50 independent FARIMA(10^5 long)(marginal distribution is heavy-tailed with tail parameter alpha = 1.6)

Method FARIMA(H=0.6) Mean Std

FARIMA(H=0.9) Mean Std

R/S 0.603 0.015 0.870 0.018

Absolute moment 0.718 0.046 0.929 0.022

ACF 0.600 0.002 0.888 0.010

Aggregate variance 0.600 0.010 0.870 0.021

Different variance 0.605 0.044 0.885 0.074

Periodogram 0.600 0.003 0.897 0.005

Box Periodogram 0.587 0.012 0.852 0.010

Higuchi’s method 0.718 0.039 0.941 0.021

Peng’s method 0.603 0.021 0.895 0.028

Wavelet - - - -

3. The Estimation of Hurst Parameter

Some of the method does not suitable for the non-normal self-similar time series.

Absolute moment method and higuchi’s method often overestimate the H for non-normal self-similar time series.

The different variance method has large estimated variance.