Wavelet-Based 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
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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)

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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

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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

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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