ikr.simlib.distributions.continuous

Class ParetoDistribution

java.lang.Object

ikr.simlib.distributions.Distribution

ikr.simlib.distributions.continuous.ContinuousDistribution

ikr.simlib.distributions.continuous.ParetoDistribution

All Implemented Interfaces:

ReflectionConstructable, ReflectionConstructable3<SimNode,Parameters,RandomNumberGenerator>

public class ParetoDistribution
extends ContinuousDistribution

Pareto Distribution

Meaning:	- Like the Weibull distribution, the Pareto distribution is often used to characterize Internet traffic because of its heavy tail. The superposition of on-off sources with heavy-tailed phases is known to produce self-similar traffic.
Parameters:	minimum value \(k > 0\) shape parameter \(\alpha > 0\) In case \(\alpha > 2\) this distribution can also be described with parameters mean value \(\mu\) and coefficient of variation \(c\): \(\alpha = 1+ \frac{\sqrt{1+c^2}}{c}\) \(k = \frac{1}{ \frac{c}{\sqrt{1+c^2}} +1 } \cdot \mu\)
PDF:	\(P(T=t) = f(t) = \frac{\alpha \cdot k^{\alpha}}{t^{\alpha +1}} \mbox{ for } t \ge k \)
DF:	\(P(T \le t) = F(t) = 1 - (\frac{k}{t})^{\alpha} \mbox{ for } t \ge k \)
Expected value:	\(E[T]= \begin{cases} \frac{\alpha}{\alpha -1} \cdot k &\mbox{ for } \alpha > 1 \\ \infty &\mbox{ for } \alpha \le 1 \end{cases} \)
Variance:	\(VAR[T]= \begin{cases} (\frac{\alpha}{\alpha -2} - (\frac{\alpha}{\alpha -1})^2) \cdot k^2 &\mbox{ for } \alpha > 2 \\ \infty &\mbox{ for } \alpha \le 2 \end{cases}\)
Coefficient of variation:	\(c_T= \frac{1}{\sqrt{(\alpha -2) \cdot \alpha}} \mbox{ for } \alpha > 2\)
Parser example:	[...].Distribution = Pareto [...].Distribution.Alpha1 = 2.2 [...].Distribution.MinValue = 1.0 or with any of the following combinations: (Mean OR MinValue) AND (CoefficientOfVariation OR Alpha), e.g., [...].Distribution.Mean = 1.83 [...].Distribution.CoefficientOfVariation = 1.51

Field Summary

Fields

Modifier and Type	Field and Description
double	alphaReciprocal
double	minValue

Fields inherited from class ikr.simlib.distributions.Distribution

rng

Fields inherited from interface ikr.simlib.parameters.reflection.ReflectionConstructable

CREATE_INSTANCE_METHOD_NAME

Constructor Summary

Constructors

Constructor and Description
ParetoDistribution(double alpha, double minValue)
ParetoDistribution(double alpha, double minValue, RandomNumberGenerator rng)

Method Summary

Methods

Modifier and Type	Method and Description
static ParetoDistribution	createInstance(SimNode ownNode, Parameters pars, RandomNumberGenerator rng) as required by ReflectionConstructable
double	next() Create random numbers

Methods inherited from class ikr.simlib.distributions.Distribution

getDefaultRNG, getRandomNumberGenerator, reset

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

alphaReciprocal

public final double alphaReciprocal

minValue

public final double minValue

Constructor Detail

ParetoDistribution

public ParetoDistribution(double alpha,
                  double minValue,
                  RandomNumberGenerator rng)

ParetoDistribution

public ParetoDistribution(double alpha,
                  double minValue)

Method Detail

createInstance

public static ParetoDistribution createInstance(SimNode ownNode,
                                Parameters pars,
                                RandomNumberGenerator rng)

as required by ReflectionConstructable

next

public double next()

Description copied from class: ContinuousDistribution

Create random numbers

Specified by:

next in class ContinuousDistribution