Wavelet-Based Network Traffic Modeling

# Wavelet-Based Network Traffic Modeling

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## Wavelet-Based Network Traffic Modeling

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1. Wavelet-BasedNetwork Traffic Modeling Carey Williamson University of Calgary

2. Introduction • Wavelets offer a powerful and flexible technique for mathematically representing network traffic at multiple time scales • Compact and concise representation of a signal using wavelet coefficients • Efficient O(N) technique for synthesizing signals as well, for N data points

3. Wavelets: Background • Wavelet transformation involves integrating a signal (continuous time or discrete) with a set of wavelet functions and scaling functions • Scaling: PHI(t) • Haar Wavelet: PSI(t)

4. Wavelets: Background • The top-level wavelet function is called the mother wavelet • The children are defined recursively using the relationship: • PHI (t) = 2 PHI(2 t - K) • PSI (t) = 2 PSI(2 t - K) J/2 J J,K J/2 J J,K where j is the (vertical) scaling level, and k is the (horizontal) translation offset, in a binary tree representation of the signal

5. Wavelets: Background • Child wavelets are narrower and taller, and cover a specific subportion of the time series • Shifted versions of the wavelet function cover other portions of the time series • Entire time series can be expressed as a sum (or integral) of scaling coefficients U and wavelet coefficients W along with these functions J,K J,K

6. Wavelets: Background • Wavelet coefficients keep track of information about the time series; in essence they keep track of the sums and/or differences between the wavelet coefficients at finer-grain time scale (plus a scaling factor) • Finest grain wavelet coefficients are derived directly from empirical time series, using C(k) = 2 Un,k n/2

7. Wavelets: Background • Coarser-grained values are computed recursively upwards using: • U = 2 (U + U ) • W = 2 (U - U ) • Topmost scaling coefficient represents mean of empirical time series • Wavelet coefficients capture the behavioural properties of the time series -1/2 J-1,K J,2K J,2K+1 -1/2 J-1,K J,2K J,2K+1

8. Wavelets: Background • Empirical time series can be exactly reconstructed using only these values (i.e., the scaling and wavelet coefficients) • Furthermore, these coefficients become decorrelated in the wavelet domain (i.e., can model arbitrary signals)

9. Wavelets: An Example • Suppose the initial empirical time series of interest has N = 8 observations in it, namely: • 17 7 12 6 10 15 8 13 (mean = 11.0) • Can construct binary tree representation of the signal and its corresponding scaling and wavelet coefficients

10. Wavelets: An Example 17 7 12 6 10 15 8 13

11. Wavelets: An Example J=0 J=1 J=2 J=3 17 7 12 6 10 15 8 13

12. Wavelets: An Example J=0 J=1 J=2 J=3 17 7 12 6 10 15 8 13 K=0 K=7

13. 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 Wavelets: An Example Compute scaling coefficients at bottom level -n/2 Un,k = 2 C(k) 17 7 12 6 10 15 8 13

14. 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 Wavelets: An Example Compute scaling coefficients at next level up -1/2 Uj-1,k = 2 (Uj,2k+Uj,2k+1) 9/2 21/4 6 25/4 17 7 12 6 10 15 8 13

15. 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 2 2 Wavelets: An Example Compute scaling coefficients at next level up 23 21 9/2 21/4 6 25/4 17 7 12 6 10 15 8 13

16. 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 2 2 Wavelets: An Example Compute scaling coefficient at top level 11 23 21 9/2 21/4 6 25/4 17 7 12 6 10 15 8 13

17. 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 2 2 Wavelets: An Example Now compute wavelet coefficients, bottom up 11 -1/2 Wj-1,k = 2 (Uj,2k-Uj,2k+1) 23 21 9/2 21/4 6 25/4 -5/4 5/2 3/2 -5/4 17 7 12 6 10 15 8 13

18. 3/2 1/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 2 2 2 2 Wavelets: An Example Now compute wavelet coefficients, bottom up 11 23 21 1 3 9/2 21/4 6 25/4 -5/4 5/2 3/2 -5/4 17 7 12 6 10 15 8 13

19. 3/2 1/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 2 2 2 2 2 2 2 2 2 2 2 2 Wavelets: An Example Now compute wavelet coefficient at top level 11 -1/2 23 21 1 3 9/2 21/4 6 25/4 -5/4 5/2 3/2 -5/4 17 7 12 6 10 15 8 13

20. 3/2 1/2 2 2 Wavelets: An Example Can reconstruct signal top-down using only the indicated information (mean and wavelet coefficients) 11 -1/2 1 3 -5/4 5/2 3/2 -5/4

21. Wavelet-Based Traffic Models • To reconstruct the time series exactly, you need to use exactly those wavelet coefficients, and the starting mean (I.e., one-to-one mapping between time series values and coefficients in the wavelet domain) • To generate something that looks like the original time series, it suffices to use Wj,k values from similar distribution

22. WIG Model • The wavelet independent Gaussian (WIG) model chooses the Wj,k’s at random from a Gaussian distribution, with a specified mean and variance at each level j of the tree (variance of the Wj,k’s at a particular level of the tree typically increases as you go down the binary tree of wavelet coefficients)

23. Wavelet-Based Traffic Modeling • In network traffic time series, the observed values are all non-negative • In wavelet terms, this constraint means the Wj,k are smaller in absolute value than the Uj,k (which themselves are always non-negative) • The WIG model does not guarantee this, and can thus generate negative values in the synthetic time series

24. Multi-Fractal Wavelet Model • The Multifractal Wavelet Model (MWM) proposed by Ribeiro et al does explicitly consider this constraint, and thus guarantees non-negative values for all observations in the generated series • Can express Wj,k = Aj,k * Uj,k where -1 <= Aj,k <= 1

25. Other Observations • For typical network traffic time series: • The mean of the Aj,k’s is zero at each level j of the binary tree of wavelet coefficients • The variance of the Aj,k’s increases as you progress down the levels of the binary tree • The Aj,k’s are uncorrelated (whether the original time series was correlated or not) • Symmetric beta distribution works well for modeling the distribution of Aj,k’s

26. Wavelet-Based Traffic Modeling • By generating random Aj,k values from a specified distribution (e.g., symmetric beta distribution), one can generate synthetic time series with desired variance (and fractal-like structure) across many time scales • Non-Gaussian marginals no problem • See example plots for LBL-TCP and Bellcore Ethernet LAN traces

27. Summary • Wavelets offer a flexible and powerful traffic modeling technique that is able to capture short-range and long-range traffic characteristics, including correlations in the time domain • Very efficient O(N) computational procedure for trace generation to generate N data points in trace