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Traffic Modeling
Approaches to construct Traffic Models

Trace-Driven:
Collect traces from the network using a sniffing tool and
utilize it directly in simulations.

Empirical Distribution:
Generate an empirical distribution from collected traces
and accordingly generate random variables to drive the
simulations.

Distribution Fitting:
Fitting collected traces to a well-known distribution. Use the
fitted distribution for both simulations and analysis .
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Trace-Driven Traffic Modeling
Packet Arrival Behavior over Time
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time
3
Empirical Distribution Traffic Modeling
Packet Arrival Behavior over Time
time
Cumulative Distribution Function (CDF)
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Generate Samples of an Empirical Model
 Generate a uniform random variable between [0,1]
 A random sample of the empirical distribution is generated
by selecting from the CDF c that corresponds to the
generated uniform distribution output.
Sample #2
Sample #1
Cumulative Distribution Function (CDF)
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Distribution Fitting Traffic Modeling
Packet Arrival Behavior over Time
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time
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Advantages and Disadvantages
Trace-Driven
Advantages
 Data is sure to be from
the correct sample.
 Practical real-life results
could be anticipated
 Sometimes there is not
enough data to figure
out the distribution
accurately
 Simulation is limited to
the results produced
from the collected data
 Data may not be
sufficient to do longDisadvantages
enough runs
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Empirical distribution
 More flexibility in terms
of data that can be
generated
 Fairly simple to deduce
from data
Fitted standard
 Same advantages as
the empirical
distribution
 Irregularities can be
smoothed out
 May have irregularities if  Can be difficult to
the collected sample is
deduce if the
not large enough
available data is
(statistical abnormalities)
limited
 The number of data
 Always a chance of
points that can be
abnormalities that
generated may be limited
were not accounted
depending on the original
for
data samples
Characterization of Distributions
Mean and Central Moments of Distributions:
Continuous Dist.
Expected Value
(Mean)
𝐸𝑋 =
Variance
𝑉𝑎𝑟 𝑋 =
Skewness
𝑆𝑘𝑒𝑤 𝑋 =
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𝐸𝑋 =
𝑥𝑓 𝑥 𝑑𝑥
𝑥−𝐸 𝑋
𝑥−𝐸 𝑋
Discrete Dist.
2
𝑓 𝑥 𝑑𝑥
𝑉𝑎𝑟 𝑋 =
3
𝑆𝑘𝑒𝑤 𝑋 =
𝑓 𝑥 𝑑𝑥
𝑥𝑃 𝑋 = 𝑥
𝑥−𝐸 𝑋
𝑥−𝐸 𝑋
2
𝑃 𝑋=𝑥
3
𝑃 𝑋=𝑥
8
Expected Value of Random Variables
 The average of the generated random samples
 If you would like to replace the whole distribution
by ONE value, the mean will have the least mean
square error
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Variance of Random Variables
 Characterizes how far does the random variable
deviate away from its mean value.
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Skewness of Random Variables
 Skewness is a measure of the asymmetry of probability
distributions
 Negative Skewness: The left tail is longer; the mass of the
distribution is concentrated on the right of the distribution.
 Positive Skewness: The right tail is longer; the mass of the
distribution is concentrated on the left of the distribution.
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Other Parameters for Continuous Distributions
 Some additional parameters of continuous distribution can
be helpful in guidance for distribution fitting
 Location Parameter (Shift Parameter):
 Specifies an abscissa (x coordinate) location point of a
distribution’s range of values,
 Often some kind of midpoint of the distribution.
 Scale Parameter:
 Determine the scale of measurement or spread of a distribution.
 Shape Parameter:
 Determine the basic form or shape of a distribution within the
family of distributions of interest.
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Location Parameter Examples
Normal Distribution
Pareto Distribution
0.4
0.7
0.35
0.6
0.3
𝝁=𝟎
𝝁=𝟐
0.5
𝜽 = 𝟐𝟎
𝜽 = 𝟏𝟎
0.25
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
-10
-8
-6
-4
𝑓 𝑥 =
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-2
0
1
2𝜋𝜎 2
2
𝑒
4
−
6
𝑥−𝝁 2
2𝜎2
8
10
0
0
10
𝑓 𝑥 =
20
30
1
𝑥−𝜽
1+𝑘
𝜎
𝜎
40
50
1
−1−𝑘
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Scale Parameter Example
 If X is a random variable with a scale parameter 1 then if
there is a random variable Y = 𝛽X then its distribution will
have scale parameter 𝛽
 The standard deviation of a normal distribution is a scale
parameter for it i. e. , 𝛽 = 𝜎
0.4
𝑿
𝝈=𝟏
0.35
0.3
0.25
0.2
0.15
𝒀 = 𝝈𝑿
𝝈=𝟐
0.1
0.05
0
-10
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-8
-6
-4
-2
0
2
4
6
8
10
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Shape Parameter Characteristics


Normal and exponential distributions do not have a shape
parameter other distributions such as beta distribution may have
two shape parameters.
A change in the shape parameter generally alter a distribution
property more fundamentally than shift or scale parameters
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Heavy-Tail Distributions


A distribution with a tail heavier than the exponential
Distributions where random variable values that are far from
the “mean” of the distribution have a non-zero probability

Pareto Principle: known as the 80-20 rule, i.e. 80% of the
effects come from 20% of the causes
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Heavy or Light Tailed?
𝑥𝑚 𝛼
,𝑥
𝑥
𝛼

Pareto Distribution: 𝐶𝐷𝐹 𝐹 𝑋 = 1 −

Pareto Survivor Function: 1-𝐶𝐷𝐹

Plot on log-log scale: A heavy tailed survivor function would be linear in log-log
domain
𝑥𝑚
𝑥
≥ 𝑥𝑚
, 𝑥 ≥ 𝑥𝑚
0
10
Exponential Dist., Mean = 30
Pareto Dist., =1.5, xm=10, Mean = 30
-2
Survivor Function (1-CDF)
10
-4
10
-6
10
Heavy Tail
-8
10
LightTail
-10
10
-2
10
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10
0
10
1
10
2
10
3
10
4
10
5
10
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Discrete Probability Distributions
Uniform (Discrete) Distribution
Binomial Distribution
Geometric Distribution
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Negative Binomial
Distribution
Hypergeometric Distribution
Continuous Probability Distributions
Uniform (Continuous)
Distribution
Triangular Distribution
Cauchy Distribution
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Normal Distribution
Exponential
Distribution
Continuous Probability Distributions
Lognormal Distribution
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Weibull Distribution
Gamma Distribution
Estimation of Parameters






Suppose that some distribution shape was deduced from the data set by any
of the methods mentioned earlier.
The data set X1,X2,…,Xn was used to deduce the distribution shape and can
be used to estimate the parameters defining the distribution completely.
There are many methods used to estimate the parameters. We will use the
maximum likelihood estimator (MLE) method.
The method can be explained as follows :
suppose we have decided that a certain discrete distribution is the closest to
the data set and that the distribution have one unknown parameter q
The Likelihood L(q) function is defined as follows :
L(q )  pq ( X 1 ) pq ( X 2 )... pq ( X n )

This is basically the joint probability mass function since the data are
assumed independent. This gives the probability of obtaining the data set as a
whole if q is the value of the unknown parameter.
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Maximum Likelihood Estimator

The MLE of the unknown value q , which is denoted by q* is defined to be the
value of q which maximizes L(q)
In the continuous case the probability mass function is substituted with the
chosen probability density function.
x/ b
Example : For an exponential distribution q  b and f b ( x)  (1 / b )e

The likelihood function will be given by :


1
1
1
L( b )  ( e  X1 / b )( e  X 2 / b )...( e  X n / b )
b
 b n exp( 
b
1
b
n
X)

b
i 1
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i
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Maximum Likelihood Estimator (cont’d)


Because most of the theoretical distributions include exponential functions it is
often easier to maximize the logarithm of the likelihood function instead of L(q)
itself.
Define the log-likelihood function as:
l ( b )  ln( L( b ))  n ln( b ) 
1
n
X

b
i 1

i
The problem reduces to maximizing the logarithmic function as the value of b
which maximizes both functions has to be the same.
dl  n 1

 2
db
b b
n
X
i 1
i
n

X
The above equation equals zero if b   i which means the sample mean
i 1 n
of the sample set.

This should be expected in the case of exponential random variables since
they are fully characterized by their means
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Maximum Likelihood Estimator (cont’d)

Suppose that the distribution chosen is a geometric distribution which is a
discrete distribution with pmf given by :
p p ( x)  p(1  p) x x  0,1,...

The likelihood function will be given by :
n
 Xi
L( p )  p (1  p ) i1
n

The log-likelihood function :
n
l ( p)  ln( L( p ))  n ln p   X i ln (1  p )
i 1

By differentiating and equating to zero
p  1 /( X (n)  1)
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