Application of Multifractals Application of Multifractals in WWW Traffic in WWW Traffic Characterization Characterization Marwan Krunz Department of Elect. & Comp. Eng. Broadband Networking Lab. University of Arizona http://www.ece.arizona.edu/ ~bnlab [email protected]
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Application of Multifractals in WWW Traffic Characterization
Application of Multifractals in WWW Traffic Characterization. Marwan Krunz Department of Elect. & Comp. Eng. Broadband Networking Lab. University of Arizona http://www.ece.arizona.edu/~bnlab [email protected]. Presentation Outline. WWW Traffic Monofractals Versus Multifractals - PowerPoint PPT Presentation
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Application of Multifractals in Application of Multifractals in WWW Traffic CharacterizationWWW Traffic Characterization
Self-Similarity in Network TrafficSelf-Similarity in Network Traffic
Self-similar traffic
Poisson traffic
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Self-Similarity … More FormallySelf-Similarity … More Formally
Consider a random process X = {X(t)} with mean ,
variance v, and ACF R(k), k = 0 , 1, …
Let X(m) be the aggregated process of X over non-
overlapping blocks of length m
X is exactly self-similar with scaling factor 0 < H < 1
if XmXandkRkR Hd
mm 1
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A process Y = {Y(t)} exhibits LRD if it is the derivative
process of a self-similar process with H > 0.5
Manifestations of LRD behavior: ACF of Y decays hyperbolically Spectral density obeys a power law near the origin: F()~c -, as 0 vVariance of the sample mean decreases more slowly than the
reciprocal of the sample size
Other Related DefinitionsOther Related Definitions
Lag
AC
F
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Multifractal ProcessesMultifractal Processes Generalizations of self-similarity, where now the H
parameter varies with scale
Wavelet construction of multifractals (Riedi et al.): Discrete wavelet transform of sequence to be modeled
Multifractals can be generated using a semi-random
cascades:
M
A1*M A2*M
A3*A1*M A4*A1*M
Ai is a symmetric random variable
If dependent semi-random cascade
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Multifractal Wavelet ModelMultifractal Wavelet Model Trace = scale coefficients at the finest time scale
For the Haar transform, the scale and wavelet coefficients are:
j 1,2k j 1,2k 1 j 1,2k j 1,2k 1j,k j,k
U U U UU and W
2 2
2 1 4 7
3/21/2 -3/21/2
14/2=7
1/21/2
-8/2=-4
Scale coefficient
Waveletcoefficient11/21/2
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Multifractal Wavelet Model Multifractal Wavelet Model
(cont.)(cont.) To generate synthetic data:
Scale coefficients at coarsest scale: U0,0 ~ N(E[U0,0],Var[U0,0])
Synthetic trace is obtained from the scale coefficients at finest scale
where Aj,k, k = 1,2,…, are iid symmetric rvs with mean zero.
Let Aj be a generic r.v. having the same CDF as Aj,k
j,k j,k j,k j,k j,k j,kj 1,2k j,k j 1,2k 1 j,k
U W 1 A U W 1 AU U and U U
2 2 2 2
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Multifractal Wavelet Model Multifractal Wavelet Model (cont.)(cont.)
Autocorrelations are controlled through the energy at scale j, i.e., E[Wj2]
To produce a synthetic trace with a desired ACF, the parameter(s) of Aj
is selected based on:
Problem: Need to compute E[Wj2] for all scales j
Large number of model parameters
2 22E W E A E Wj 1 0j 1 2 and E A0j 22 22E W E UE A 1 E A 0j j j 1
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Goal: Reduce the complexity of the original model
Outline of modified model: Take Aj to be a triangular rv in the range[-cj, cj] for all j Define the aggregated sequence {Xn
(m) : n = 1, 2, …}
Relate E[(Xn(m))2] to E[Uj
2] and, thus, to E[Aj2]
Aggregation level 2m represents the scale j-1
Express cj-1 c(2m) in terms of E[(Xn(m))2] and E[(Xn
(2m))2]
Modified Multifractal ModelModified Multifractal Model
nm
mn i
i nm m 1
X X ,n 1,2, , N / m,m 1,2,4, , N
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Modified Multifractal Model (cont.)Modified Multifractal Model (cont.) Relate E[(Xn
(m))2] to the mean (), variance (v), and ACF (k: k = 1,2,…) of the original trace:
Thus, cj,j = 1, 2, …,is expressed in terms of , v, and k: k = 1,2,…
For the ACF, we use the general form:
g(k) is taken to be k or log(k+1)
model is specified using 4 parameters
m2(m) 2 2
n kk 1
E X mv 2v m k m
exp( ( )), 0,1,...nk g k k
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Outline of Traffic GenerationOutline of Traffic Generation
1. Extract empirical (scaled) stack distancesa. Start with an empty stack (to avoid initial ordering problem)b. Process trace in the reverse directionc. Record stack depth only for objects already in the stackd. Reverse the extracted stack distance stringe. Normalize stack distances by their empirical averages
2. Generate synthetic stack distance stringa. Compute parameters for multifractal modelb. Generate a synthetic (scaled) stack distance stringc. Scale back stack distances
3. Generate URL traces while enforcing popularity profile
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Traffic Generation ExampleTraffic Generation Example