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WaveletBased Network Traffic Modeling

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... observed values are all non-negative ... Symmetric beta distribution works well for modeling the distribution of Aj,k's ... Non-Gaussian marginals no problem ... – PowerPoint PPT presentation

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Title: WaveletBased Network Traffic Modeling


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,2K1
-1/2
J-1,K
J,2K
J,2K1
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
J0
J1
J2
J3
17
7
12
6
10
15
8
13
12
Wavelets An Example
J0
J1
J2
J3
17
7
12
6
10
15
8
13
K0
K7
13
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
Wavelets An Example
Compute scaling coefficients at next level up
-1/2
Uj-1,k 2 (Uj,2kUj,2k1)
9/2
21/4
6
25/4
17
7
12
6
10
15
8
13
15
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
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
Wavelets An Example
Now compute wavelet coefficients, bottom up
11
-1/2
Wj-1,k 2 (Uj,2k-Uj,2k1)
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
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
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
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,ks at random from a Gaussian
    distribution, with a specified mean and variance
    at each level j of the tree (variance of the
    Wj,ks 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
    lt Aj,k lt 1

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

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
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