1. Presented by Chun Zhang 2/14/2003
On the Self-Similar Nature of Ethernet Traffic Will E. Leland, Walter Willinger and Daniel V. Wilson BELLCORE Murad S. Taqqu BU Appeared on SIGCOM 1993
Presented by Chun Zhang
2/14/2003
CMPSCI 791Y : Measurement Seminar

2. Overview
Self Similarity
Ethernet Traffic is Self-Similar

3. Section 1: Self-Similarity

4. Why is Self-Similarity Important?
In this paper, Ethernet traffic has been identified as being self-similar.
Models like Poisson are not able to capture the self-similarity property.
This leads to inaccurate performance evaluation

5. Intuition of Self-Similarity
Something “feels the same” regardless of scale (also called fractals)

6. Self-Similarity in Traffic Measurement (Ⅰ) Traffic Measurement

7. Self-Similarity in Traffic Measurement (Ⅱ) Network Traffic

8. Self-Similarity in Traffic Measurement (Ⅲ) Self-Similarity
X = (Xt : t = 0, 1, 2, ….) is covariance stationary process
(i.e. Cov(Xt,Xt+k) does not depend on t for all k)
Mean , variance 2
X(m)={Xk(m)} where elements are average over non-overlapping blocks of m
Suppose that r(k)  k-β, 0<β<1
X is [asymptotically] second-order self-similar with Hurst parameter H (= 1- β/2) if for all m,
Var(X(m) ) = 2 m-β , and
r (m) (k) = r(k), k0 [ r (m) (k)  r(k), m∞  ]

9. Properties of Self-Similarity
Var(X(m) ) (= 2 m-β ) decreases more slowly (than m –1)
r(k) decreases hyperbolically (not exponentially) so that
kr(k) =  (long range dependence)
The spectral density [discrete time Fourier Transform of r(k)] f(λ) cλ-(1- β), as λ0 (not bounded)

10. Modeling Self-Similarity
Fractional Gaussian noise (FGN)
Gaussian process with mean , variance 2, and
Autocorrelation function r(k)=(|k+1|2H-|k|2H+|k-1|2H), k>0
Exactly second-order self-similar with 0.5<H<1
Fractional ARIMA(p,d,q)
Asymptotically second-order self-similar with H=d+0.5 where 0<d<0.5
Discrete time M/G/ input model
Service time X given by heavy tail distribution (i.e. E[x]< , 2= )
Example : Pareto distribution P(X>k) k-α, 1< α<2
N = {Nt ,t=1,2,…} is self-similar with H=(3- α)/2 where Nt denotes # of members being serviced at time t

11. Inference for Self-Similar Processes
Time domain analysis based on R/S statistic
Variance analysis based on the aggregated process X(m)
Reminds Var(X(m) ) = 2 m-β, plot log(Var(X(m) )) against log m
Frequency domain analysis
Estimate PSD f(λ) using discrete time Fourier Transform
Reminds f(λ) cλ-(1- β), as λ0, plot log(f(λ)) against log λ
Provides confidence intervals when combining Whittle’s MLE approach and the aggregation method

12. Section 2: Ethernet Traffic is Self-Similar

13. Data Collection Ethernet traffic of a large lab at Bellcore
100µs time resolution
Packet length, interface status, header info
Mostly IP and NFS
Four data sets over three year period
Over 100M packets in traces

14. Plots Showing Self-Similarity (Ⅰ)
Estimate H  0.8

15. Plots Showing Self-Similarity (Ⅱ)
High Traffic
5.0%-30.7%
Mid Traffic
3.4%-18.4%
Low Traffic
1.3%-10.4%
Higher Traffic, Higher H

16. H : A Function of Network Utilization
Observation shows “contrary to Poisson”
Network Utilization H
As we shall see shortly, H measures traffic burstiness
As number of Ethernet users increases, the resulting
aggregate traffic becomes burstier instead of smoother

17. H : Measuring “Burstiness”
Intuitive explanation using M/G/ Model
As α 1, service time is more variable, easier to generate burst
Increasing H !
Wrong way to measure “burstiness” of self-similar process
Peak-to-mean ratio
Coefficient of variation (for interarrival times)

18. Summary Ethernet LAN traffic is statistically self-similar
H : the degree of self-similarity
H : a function of utilization
H : a measure of “burstiness”
Models like Poisson are not able to capture self-similarity

19. Discussions How to explain self-similarity ?
Heavy tailed file sizes
How this would impact existing performance?
Limited effectiveness of buffering
Effectiveness of FEC
How to adapt to self-similarity?
Prediction
Adaptive FEC

20. Thanks !