MARTE — Section: 8.3.3.1

Proposition to extend the probability distributions functions list with (used in AnyLogic) : - geometric (double p) The Geometric distribution is a discrete distribution bounded at 0 and unbounded on the high side. It is a special case of the Negative Binomial distribution. In particular, it is the direct discrete analog for the continuous Exponential distribution. The Geometric distribution has no history dependence, its probability at any value being independent of a shift along the axis. - laplace (double phi, double beta) The Laplace distribution, sometimes called the double exponential distribution, is an unbounded continuous distribution that has a very sharp central peak, located at theta. The distribution scales with phi. - chi squared (double nu, double min) The Chi Squared is a continuous distribution bounded on the lower side. Note that the Chi Squared distribution is a subset of the Gamma distribution with beta=2 and alpha=nµ/2. Like the Gamma distribution, it has three distinct regions. For nµ=2, the Chi Squared distribution reduces to the Exponential distribution, starting at a finite value at minimum x and decreasing monotonically thereafter. For nµ<2, the Chi Squared distribution tends to infinity at minimum x and decreases monotonically for increasing x. For nµ>2, the Chi Squared distribution is 0 at minimum x, peaks at a value that depends on nµ, decreasing monotonically thereafter. - rayleigh (double sigma) The Rayleigh distribution is a continuous distribution bounded on the lower side. It is a special case of the Weibull distribution with alpha =2 and beta/sqrt(2) =sigma. Because of the fixed shape parameter, the Rayleigh distribution does not change shape although it can be scaled. - weibull (double alpha, double beta, double min) The Weibull distribution is a continuous distribution bounded on the lower side. Because it provides one of the limiting distributions for extreme values, it is also referred to as the Frechet distribution and the Weibull-Gnedenko distribution. - logistic (double beta, double alpha) The Logistic distribution is an unbounded continuous distribution which is symmetrical about its mean [and shift parameter], alpha. The shape of the Logistic distribution is very much like the Normal distribution, except that the Logistic distribution has broader tails. - pareto (doubla alpha, double min) The Pareto distribution is a continuous distribution bounded on the lower side. It has a finite value at the minimum x and decreases monotonically for increasing x. A Pareto random variable is the exponential of an Exponential random variable, and possesses many of the same characteristics. - triangular (double min, double max, double mode) The Triangular distribution is often used when no or little data is available; it is rarely an accurate representation of a data set. However, it is employed as the functional form of regions for fuzzy logic due to its ease of use. - cauchy (doubla lambda, double theta) The Cauchy distribution is an unbounded continuous distribution that has a sharp central peak but significantly broad tails. The tails are much heavier than the tails of the Normal distribution. - beta (double p, double q, double min, double max) The Beta distribution is a continuous distribution that has both upper and lower finite bounds. Because many real situations can be bounded in this way, the Beta distribution can be used empirically to estimate the actual distribution before much data is available. Even when data is available, the Beta distribution should fit most data in a reasonable fashion, although it may not be the best fit. The Uniform distribution is a special case of the Beta distribution with p, q = 1. - lognormal (double mu, double sigma, double min) The Lognormal distribution is a continuous distribution bounded on the lower side. It is always 0 at minimum x, rising to a peak that depends on both mu and sigma, then decreasing monotonically for increasing x. - erlang (double beta, int m, double min) The Erlang distribution is a continuous distribution bounded on the lower side. It is a special case of the Gamma distribution where the parameter, m, is restricted to a positive integer. As such, the Erlang distribution has no region where F tends to infinity at the minimum value of x [m<1], but does have a special case at m=1, where it reduces to the Exponential distribution. - negativeBinomial (double p, double n) The Negative Binomial distribution is a discrete distribution bounded on the low side at 0 and unbounded on the high side. The Negative Binomial distribution reduces to the Geometric Distribution for k = 1. The Negative Binomial distribution gives the total number of trials, x, to get k events (failures...), each with the constant probability, p, of occurring. - logarithmic (double beta) The Logarithmic distribution is a discrete distribution bounded by [1,...]. Theta is related to the sample size and the mean. - hypergeometric (int ss, int dn, int ps) The Hypergeometric distribution is a discrete distribution bounded by [0,s]. It describes the number of defects, x, in a sample of size s from a population of size N which has m total defects. The ratio of m/N = p is sometimes used rather than m to describe the probability of a defect. Note that defects may be interpreted as successes, in which case x is the number of failures until (s-x) successes. The sample is taken without replacement.

We need to be sure that the new distributions functions are really necessary in
MARTE. Note that we do not attempt in MARTE to define all the existing
distribution functions but the required in common practice.
After evaluating some performance analysis and simulation tools, we consider
that the following one are highly decided, and propose to include them in the
MARTE library. geometric (real p). The Geometric distribution is a discrete distribution
bounded at 0 and unbounded on the high side.

• triangular (real min, real max, real mode). The Triangular distribution is
  often used when no or little data is available; it is rarely an accurate
  representation of a data set.
• logarithmic (real theta). The Logarithmic distribution is a discrete
  distribution bounded by [1,...]. Theta is related to the sample size and the
  mean.
  Further distribution functions can be added at library level.
  Issue Dependency Warning: Note that this issue affects Issue 12561, which
  clarifies the mechanism to specify probability distribution expressions. Issue
  12561 depends on this issue, but the reverse case is not true.