1. Chen Chu South China University of Technology

2. 1. Self-Similar process and Multi-fractal process There 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 -H Y(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.

3. 1. Self-Similar process and Multi-fractal process Definition 2: A wide-sense stationary sequence X(i). For each m = 1, 2, 3,..., Let X k (m) ＝ 1/m(X km-m+1 ＋ … ＋ X km ) ， 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.

4. 1. Self-Similar process and Multi-fractal process Definition 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.

5. 1. Self-Similar process and Multi-fractal process Self-similar processCharacteristics Definition 1FBM: self-similar, normal distribution, stationary increments Definition 2FGN: 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 3FGN: FARIMA: normal distribution. For non-normal distributional FARIMA, we get different H with different q. It is Multi-fractal process according to definition II.

6. 1. Self-Similar process and Multi-fractal process Multi-fractal processCharacteristics Definition IMFBM: non-stationary, non-stationary increments Definition II(non-normal FARIMA): self-similar, stationary, long-range dependent, slowly decaying variances, Hurst effect, non-normal distribution

7. 2. The Characteristics of Network Traffic 1): 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.

8. 2. The Characteristics of Network Traffic MAWI IPv6 WIDE backbone. The scaling phenomena exist during 100ms~100s. For time scale less than 100ms, there is no self-similar exist.

9. 2. The Characteristics of Network Traffic 2): The marginal distribution of network traffic is not normal. But as the time scale increase, it becomes normal.

10. 2. The Characteristics of Network Traffic 3): The long-dependence of the traffic.

11. 2. The Characteristics of Network Traffic BC traffic Aug89MAWI IPv6 traffic environmentLAN, IPv4WAN, Backbone, IPv6 Self-similarityForm 10ms to 100sForm 100ms to 100s There is no self-similarity when time scale less than 100ms. DistributionAlmost normalAsymptotically normal Long-dependenceThe 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.

12. 2. The Characteristics of Network Traffic Conclusion: Larger scales (time scale larger than 10ms or even 100ms) self-similarity long-dependence nearly normal distribution. Small scales (time scale less than 100ms) any distribution (usually lognormal or heavy-tail) not self-similar short-dependent

13. 2. The Characteristics of Network Traffic Models 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.

14. 3. The Estimation of Hurst Parameter The 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) MethodFGN(H=0.5) Mean Std FGN(H=0.9) Mean Std R/S0.5150.0150.8680.021 Absolute moment0.5000.0100.8680.015 ACF0.5000.0030.8880.005 Aggregate variance0.5000.0090.8680.014 Different variance0.5010.0120.8980.010 Periodogram0.5010.0060.9050.006 Box Periodogram0.4970.0110.8560.011 Higuchi’s method0.5010.0090.8940.030 Peng’s method0.5010.0100.9000.013 Wavelet0.5030.0030.9170.004 Whittle0.5000.0020.9000.002

15. 3. The Estimation of Hurst Parameter Estimation of H with 50 independent FARIMA(10^5 long) (marginal distribution is heavy-tailed with tail parameter alpha = 1.8) MethodFARIMA(H=0.6) Mean Std FARIMA(H=0.9) Mean Std R/S0.6090.0120.8710.018 Absolute moment0.6530.0030.9070.025 ACF0.6000.0180.8880.005 Aggregate variance0.5980.0100.8700.015 Different variance0.6050.0340.8810.061 Periodogram0.6010.0060.8970.006 Box Periodogram0.5850.0120.8540.013 Higuchi’s method0.6530.0170.9200.027 Peng’s method0.5980.0140.8930.020 Wavelet----

16. 3. The Estimation of Hurst Parameter Estimation of H with 50 independent FARIMA(10^5 long) (marginal distribution is heavy-tailed with tail parameter alpha = 1.6) MethodFARIMA(H=0.6) Mean Std FARIMA(H=0.9) Mean Std R/S0.6030.0150.8700.018 Absolute moment0.7180.0460.9290.022 ACF0.6000.0020.8880.010 Aggregate variance0.6000.0100.8700.021 Different variance0.6050.0440.8850.074 Periodogram0.6000.0030.8970.005 Box Periodogram0.5870.0120.8520.010 Higuchi’s method0.7180.0390.9410.021 Peng’s method0.6030.0210.8950.028 Wavelet----

17. 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.