Application of Multifractals in Application of Multifractals in WWW Traffic CharacterizationWWW Traffic Characterization

Marwan KrunzDepartment of Elect. & Comp. Eng.

Broadband Networking Lab.University of Arizona

http://www.ece.arizona.edu/~bnlab

krunz@ece.arizona.edu

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Presentation OutlinePresentation Outline

WWW Traffic

Monofractals Versus Multifractals

Proposed Model

Simulation Results

Ongoing Research

Other BNL Projects

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WWW TrafficWWW Traffic

What do we mean by WWW traffic? Sequence of requests for file objects at a server

Why do we want to model it? Capacity planning & resource dimensioning Design of caching & prefetching schemes

What traffic properties to capture? Popularity Temporal locality Spatial locality

C CC

C

C

C Web server

Internet

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Temporal LocalityTemporal Locality

Closeness in time between references to the same object

Often measured using the stack distance string

A B A C B D A B A C B D

time

D

B

A

C

2

D

C

A

B

2

C

D

B

A

C

D

A

B

C

B

A

D

3

D

A

B

C

4

C

D

A

B

4

B

C

D

A

33 2

D

C

A

B

Sta

ck

stack distance string1

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Temporal Locality (cont.)Temporal Locality (cont.) Temporal locality is often represented by the

marginal distribution of the stack distance string Approximately lognormal

Sources of temporal locality: “Long-term” popularity of objects Temporal correlations between requests to same object

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Temporal Locality (cont.)Temporal Locality (cont.)

Need to differentiate between the two sources, since Long-term popularity suggests the use of long-term

frequency information in caching (LFU) Temporal correlations suggest the use of short-term

residency information in caching (LRU)

Solutions: Have several, popularity-based stack-distance models Use a scaled version of the stack distance string

[Cherkasova & Ciardo, 2000] Stack distances normalized by their mean stack distance

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Example – CLARKNET TraceExample – CLARKNET Trace

Popularity (# of requests)

Mea

n s

tack

dis

tan

ce

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Spatial LocalitySpatial Locality

Correlations between requests to different files

Can be captured through the autocorrelation function

(ACF) of the (scaled) stack distance string

Empirical ACF exhibits a slowly decaying behavior;

an indication of long-range dependence (LRD)

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ExampleExample

A B F G E D A H B F E D

time

67

7

Stack distance: 6,7,7,…

high autocorrelation value at lag 1 of the stack distance string

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ACF – Calgary TraceACF – Calgary Trace

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240

Lag

ACF

Real

Fitted

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How to Simultaneously Capture How to Simultaneously Capture Temporal and Spatial LocalitiesTemporal and Spatial Localities

Previous approach: Self-similar model (Crovella et al.) Start with a F-ARIMA with a desired H-parameter

Transform the Gaussian distribution of the F-ARIMA

process into a lognormal distribution

Problems: H-parameter characterizes only the long-term correlation

behavior

Transformation is nonlinear (hence, it does NOT preserve the

overall structure of the ACF)

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Impact of Transformation Impact of Transformation

Lag (in stack distances)

Au

toco

rrel

atio

ns

F-ARIMA (after transformation)

Real

F-ARIMA (before transformation)

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Monofractals (Self-Similarity)Monofractals (Self-Similarity)

Example from geometry: The Sierpinski gasket

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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

, , , ,( ) ( ) ( )J k J k j k j kk j J k

X t U t W t

Scale coefficient atscale J and time 2Jk

Shifted and translated scale function

Wavelet coefficient atscale j and time 2jk

Shifted and translated wavelet function

Coarsestscale

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Multifractal Processes (cont.)Multifractal Processes (cont.)

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

Trace length=12

popularity profile:

frA=4/12

frB=4/12

frC=2/12

frD=2/12

D

C

A

B

synthetic traffic

A

D

C

A

B

B

Scaled back synthetic stack distance string

2434432 3

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Simulation ResultsSimulation Results

RealMultifractal model

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Simulation Results (cont.)Simulation Results (cont.)

RealMultifractal model

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Simulation Results (cont.)Simulation Results (cont.)

Statistics Real MF LRD No spatial loc.

0.954 0.937 0.905 0.806

1.032 1.164 1.428 1.918

10.13 0.118 0.06 0.0

50.076 0.075 0.02 0.0

250.039 0.039 0.001 0.0

90th percentile 2.233 2.21 2.16 1.67

98th percentile 3.976 4.23 5.05 4.98

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Ongoing & Future WorkOngoing & Future Work

Online traffic forecasting using multifractal model

Incorporation of traffic forecasting in design of

prefetching strategies

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Modeling of Prefetching SystemsModeling of Prefetching Systems

Goal: Provide a theoretical model to analyze the

performance of generic prefetching systems

Limitations of existing works: Many are mainly focused on the prediction aspect only Performance is often studied via simulations under very

specific setups (e.g., given network topology) Few analytical models, but which are overly simplistic

(e.g., ignore client behavior, TCP dynamics, etc.)

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FrameworkFramework

Local cache

One TCP connection for both, demand fetching and prefetching

Two separate TCP connections

Prefetching cache: Small portion of the local cache

Server

Client: ON/OFF source

Predictor: I suggest you prefetch documents:D1,D2,….,Dk since I think they will be requestedsoon with probabilities: P1, P2, …., Pk.

Should I onlyuse thinking time for prefetching?

An access toa prefetched document movesthe document to the local cache.This helps in Studying the performance ofprefetching in isolation from caching.

OR