1. 1 Self Similar Traffic

2. 2 Self Similarity The idea is that something looks the same when viewed from different degrees of “magnification” or different scales on a dimension, such as the time dimension. It’s a unifying concept underlying fractals, chaos, power laws, and a common attribute in many laws of nature and in phenomena in the world around us.

3. 3 Cantor Each left portion in a step is a full replica of the preceding step

4. 4 Fractals

5. 5 What is a Fractal? Exhibits self-similarity Based on recursive algorithms Unique dimensionality Scale independent

6. 6 Network Traffic in the Real World For years, traffic was assumed to be based on Poisson. It is now known that this traffic has a self similar pattern. Characterized by burstiness.

7. 7 Self Similarity of Ethernet Traffic Seminal paper by W. Leland et al published in 1993, examined Ethernet traffic between 1989 and 1992, gathering 4 sets of data, each lasting 20 to 40 hours, with a resolution of 20 microseconds. Paper shattered the illusion of Poison distribution being adequate for traffic analysis. Proved Ethernet traffic is self similar with a Hurst factor of H = 0.9 0 < H <1 ; the higher H, the more self similar the pattern

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9. 9 Self-similarity manifests itself in several equivalent fashions: –Slowly decaying variance –Long range dependence –Non-degenerate autocorrelations –Hurst effect Self-similarity: manifestations

10. 10 Definition of Self-Similarity Self-similar processes are the simplest way to model processes with long-range dependence – correlations that persist (do not degenerate) across large time scales The autocorrelation function r(k) of a process (statistical measure of the relationship, if any, between a random variable and itself, at different time lags)with long-range dependence is not summable: –  r(k) = inf. – r(k)  k -  as k  inf. for 0 <  < 1 Autocorrelation function follows a power law Slower decay than exponential process –Power spectrum is hyperbolic rising to inf. at freq. 0 –If  r(k) < inf. then you have short-range dependence

11. 11 Self-Similarity contd. Consider a zero-mean stationary time series X = (X t ;t = 1,2,3,…), we define the m-aggregated series X (m) = (X k (m) ;k = 1,2,3,…) by summing X over blocks of size m. We say X is H-self-similar if for all positive m, X (m) has the same distribution as X rescaled by m H. If X is H-self-similar, it has the same autocorrelation function r(k) as the series X (m) for all m. This is actually distributional self- similarity. Degree of self-similarity is expressed as the speed of decay of series autocorrelation function using the Hurst parameter –H = 1 -  /2 –For SS series with LRD, ½ < H < 1 –Degree of SS and LRD increases as H  1

12. 12 Graphical Tests for Self-Similarity Variance-time plots –Relies on slowly decaying variance of self-similar series –The variance of X (m) is plotted versus m on log-log plot –Slope (-  greater than –1 is indicative of SS R/S plots –Relies on rescaled range (R/S)statistic growing like a power law with H as a function of number of points n plotted. –The plot of R/S versus n on log-log has slope which estimates H Periodogram plot –Relies on the slope of the power spectrum of the series as frequency approaches zero –The periodogram slope is a straight line with slope  – 1 close to the origin

13. 13 Graphical test examples – VT plot

14. 14 Graphical test example – R/S plot

15. 15 How Inaccurate Are Older Models?