| 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 \muand 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. | | | | | | | | | | | | | | |