# Package ikr.simlib.distributions.continuous

Continuous Distributions
The classes derived from ContinuousDistribution override the abstract method next(), which returns a double value.

See: Description

• Class Summary
Class Description
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).
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 
• Enum Summary
Enum Description
GeneralDistribution.SubType
• ## Package ikr.simlib.distributions.continuous Description

Continuous Distributions
The classes derived from ContinuousDistribution override the abstract method next(), which returns a double value.