| BetaDistribution |
Beta Distribution
| Meaning:
Distribution of a random variable \(T= \frac{Z_1}{Z_1+Z_2}\), if \(Z_1\) and
\(Z_2\) are gamma distributed with the parameters \(\alpha_1\) and \(\beta\)
as well as \(\alpha_2\) and \(\beta\).
|
| ConstantDistribution |
Deterministic Value
| Meaning:
A constant value \(d\) is returned
| Parameters:
constant ("mean") value \(d\)
| PDF:
\(P(T=t) = f(t) = \delta(t-d) \)
| DF:
\(P(T \le t) = F(t) = \sigma(t-d) = \begin{cases} 0 &\mbox{for } t < d \\
1 & \mbox{else } \end{cases} \)
| Expected value:
\(E[T]= d \)
| Variance:
\(VAR[T]= 0\)
| Coefficient of variation:
\(c_T= 0\)
| LST:
\(\phi(s) = exp(-sd)\)
| Parser example:
[...].Distribution = Constant
[...].Distribution.Mean = 1.7
|
| ContinuousDistribution |
ContinuousDistributions are distributions that return floating
point values.
|
| CoxianDistribution |
Coxian Phase Model
Coxian Phase Model
| Meaning:
Distribution according to the Coxian phase model: Serial switching of a
selection of one of \(k\) phases each with a negative-exponentially
distributed phase duration period (parameter \(\lambda_i\)), whereby after
each phase the system is exited with the probability \(q_i\).
|
| EmpiricalDistribution |
EmpiricalDistribution allows to read empirical distributions
obtained e.g.from measurements into a simulation program.
|
| ErlangDistribution |
Erlang k Distribution
Erlang k distribution
| Meaning:
Distribution for the sum of \(k\) random variables that are each
negative-exponentially distributed with the parameter \(\lambda\) (serial
switch in the phase model).
|
| GammaDistribution |
Gamma Distribution
| Meaning:
The gamma distribution may, e.g., be applied to characterize video
traffic.
|
| GeneralDistribution |
General Distribution
Meaning:
A distribution with a certain mean value \(\mu\)and coefficient of
variation \(c\) is generated by linking two phases (phase model)
case I \((c=0)\):
Only one phase with a constant distribution (mean value \(\mu\)).
|
| HyperExpDistribution |
Hyperexponential Distribution to the Order of k
Hyperexponential distribution to the order of k
| Meaning:
Selecting one of \(k\) random variables that are negative-exponentially
distributed with the individual parameters \(\lambda_i\) and the
probabilities \(p_i\) (parallel switching in the phase model).
|
| HypoExpDistribution |
Hypoexponential Distribution to the Order of k
Hypoexponential distribution to the order of k
| Meaning:
Generalization of the Erlang k distribution (serial switching of \(k\)
phases with negative-exponentially distributed durations with individual
parameters \(\lambda_i\) in the phase model).
|
| JakesDistribution |
Jakes Distribution
Meaning:
The class
JakesDistribution
implementes a continuous distribution which delivers
random variables in accordance with Jake's Doppler power density spectrum.
|
| LognormalDistribution |
Lognormal Distribution
| Meaning:
Distribution of \(T = exp(Z)\), if \(Z\) is normally distributed with the
parameters \(\mu\) and \(\sigma^2\).
|
| NegExpDistribution |
Negative Exponential Distributione
| Meaning:
Time distance between two events (e.g., arrival, process end,...) in a
Markovian process with the average rate \(\lambda\)
| Parameters:
mean value \(m\) or rate \(\lambda = \frac{1}{m}\)
| PDF:
\(P(T=t) = f(t) = \lambda \cdot exp(-\lambda t) = \frac{1}{m} \cdot
exp(-\frac{t}{m}) \)
| DF:
\(P(T \le t) = F(t) = 1-exp(-\lambda t) = 1-exp(-\frac{1}{m}) \)
| Expected value:
\(E[T]= \frac{1}{\lambda} = m \)
| Variance:
\(VAR[T]= \frac{1}{\lambda^2} = m^2 \)
| Coefficient of variation:
\(c_T= 1\)
| LST:
\(\phi(s) = \frac{\lambda}{\lambda + s} = \frac{1}{1 + ms}\)
| Parser example:
[...].Distribution = NegExp
[...].Distribution.Mean = 3.6
|
| NormalDistribution |
Normal Distribution
| Meaning:
Limit distribution of the sum of many independent random variables with
arbitrary statistical characteristics, if the contribution of a single random
variable remains neglectably small.
| Parameters:
mean value \(\mu\)
standart deviation \(\sigma > 0\)
| PDF:
\(P(T=t) = f(t) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot exp( -\frac{(t-
\mu)^2}{2\sigma^2} )\)
| DF:
\(P(T \le t) = F(t) = \frac{1}{2} \cdot erf(\frac{t-\mu}{\sqrt{2}
\sigma}) \mbox{ with } erf(x) = \frac{2}{\sqrt{\pi}} \cdot \int_0^x \!
exp(-y^2) \, dy \)
| Expected value:
\(E[T]= \mu \)
| Variance:
\(VAR[T]= \sigma^2\)
| Coefficient of variation:
\(c_T= \frac{\sigma}{\mu}\)
| LST:
\(\phi(s) = exp(-\mu s + \frac{(s \sigma)^2}{2})\)
| Parser example:
[...].Distribution = Normal
[...].Distribution.Mean = 6.3
[...].Distribution.StandardDeviation = 1.5
|
| ParetoDistribution |
Pareto Distribution
| Meaning:
- Like the Weibull distribution, the Pareto distribution is often used to
characterize Internet traffic because of its heavy tail.
|
| RayleighDistribution |
|
| UniformDistribution |
Uniform Distribution
| Meaning:
All (continuous) values \(t\) in the interval \(b_l < t < b_u\) appear
with the same probability
| Parameters:
lower limit \(b_l\)
upper limit \(b_u > b_l \)
| PDF:
\(P(T=t) = f(t) = \begin{cases} \frac{1}{b_u -b_l} &\mbox{for } b_l \le t
\le b_u \\ 0 &\mbox{else } \end{cases}\)
| DF:
\(P(T \le t) = F(t) = \begin{cases} 0 &\mbox{for } t < b_l \\
\frac{t-b_l}{b_u-b_l} &\mbox{for } b_l \le t < b_u \\ 1 &\mbox{for } t \ge
b_l \end{cases} \)
| Expected value:
\(E[T]= \frac{b_l + b_u}{2}\)
| Variance:
\(VAR[T]= \frac{(b_u - b_l)^2}{12}\)
| Coefficient of variation:
\(c_T= \frac{1}{\sqrt{3}} \cdot \frac{b_u-b_l}{b_u+b_l}\)
| LST:
\(\phi(s) = \frac{1}{b_u-b_l} \cdot \frac{exp(-b_l s) - exp(-b_u s)}{s}\)
| Parser example:
[...].Distribution = Uniform
[...].Distribution.LowerBound = 1.5
[...].Distribution.UpperBound = 13.5
|
| WeibullDistribution |
Weibull Distribution
| Meaning:
The Weibull distribution is often used to model internet traffic because
of its heavy tail.
| Parameters:
shape parameter \(\alpha > 0\)
scale parameter \(\beta > 0\)
| PDF:
\(P(T=t) = f(t) = \alpha \cdot \beta^{-\alpha} \cdot t^{\alpha -1} \cdot
exp(-(\frac{t}{\beta})^{\alpha}) \mbox{ for } t>0 \)
| DF:
\(P(T \le t) = F(t) = 1- exp(-(\frac{t}{\beta})^{\alpha}) \mbox{ for }
t>0\)
| Expected value:
\(E[T]= \frac{\beta}{\alpha} \cdot \Gamma(\frac{1}{\alpha}\), whereas
\(\Gamma(x)\) is the gamma function
| Variance:
\(VAR[T]= \frac{\beta^2}{\alpha} \cdot \left\{ 2\Gamma(\frac{2}{\alpha})
- \frac{1}{\alpha} \cdot \Gamma(\frac{1}{\alpha})^2 \right\} \)
| Coefficient of variation:
\(c_T=
\sqrt{\frac{2\alpha\Gamma(\frac{2}{\alpha})}{(\Gamma(\frac{1}{\alpha}))^2}
-1}\)
| Parser example:
[...].Distribution = Weibull
[...].Distribution.Alpha = 0.3
[...].Distribution.Beta = 0.03
|
Package ikr.simlib.distributions.continuous Description
Continuous Distributions
The classes derived from
ContinuousDistribution override
the abstract method next(), which returns a double
value. | | | | | | | | | | | | | | |