Probability distributions

The generator and analysis classes use the following probability distributions as described in [Watkins 93] and [Kalvelagen 90]. Some use the function, which is defined as :

The distributions are :

• the uniform distribution, this distribution represents the situation whereby a random number can take values within a finite range with equal probability. It is characterized by the following function :

The method in the library takes the upper and lower limit as parameters.

• the exponential distribution, takes the non-negative mean as parameter. Its function :

This function is used to model purely random events such as the time between arrivals of a customer.

• the hyperexponential distribution, takes the non-negative means and of two exponential distributions as parameters. It then generates a variate from the first distribution with chance p and a variate of the second with chance 1-p. Its function :

• the normal distribution, the first parameter to this distribution is the mean , the second is the standard deviation . It is characterized by the following probability distribution :

This function is used for the modeling of the additive effect of several independent trials.

• the lognormal distribution, It is calculated as the logarithm of the normal distribution with as mean and as standard deviation. Its probability function :

• the erlang distribution. It is used to model service times in a queuing system that has n servers where each server has a mean service time of . The overall service time of this system is given by the erlang distribution. It is characterized by the following distribution :

• the gamma distribution, this distribution is parameterized by two values and , both greater then 0. It has the following probability distribution :

• the poisson distribution, gets the mean as parameter. This function is used to model the number of occurrences, such as the number of component failures, over a given period of time. It is characterized by the following distribution :

• the geometric distribution, this distribution is characterized by p, the chance on success. It is used to model the number of occurrences between two significant events as the number of failures to acquire a particular resource until the first success. Its distribution :

• the hypergeometric distribution. Suppose that a sample of size n is to be randomly chosen (without replacements) from an urn containing m balls of which mp are white and m(1-p) are black. The chance of x white balls selected follows the hypergeometric distribution and is equal to :

• the weibull distribution, it gets and as parameters, both greater then 0. It is used for reliability measures such as the lifetime of components. It is characterized by the following distribution function :

• the binomial distribution, gets as parameters p, the chance on a success and n, the number of trials. It is used to model the number of successes in n independent Bernoulli trials. It has the following probability distribution :

• the negative binomial distribution, it is parameterized by p, the chance on a success, and n, the number of successes. The negative binomial distribution returns the number of failures before the success. You can think of the number of retransmissions of a file. The negative binomial distribution is characterized by the following function :

• the beta distribution, this probability function is characterized by the two parameters and . It is used to model random proportions as the fraction of customers that could not be served. It has the following distribution :

• the laplace distribution. The laplace distribution looks like the exponential distribution with a mirror image in the y-axis. It has the following probability function :

• the distribution, with n, the degrees of freedom as parameter. It has as probability distribution function :

The distribution is a special case of the gamma distribution with and .

• the student's t distribution, it also gets the degrees of freedom as parameter. If n is large enough (), it can be approximated by the standard normal distribution. It has as function :

• the F distribution, returns a value of the F distribution, which is often used for testing variances. The distribution takes two positive integers as arguments, defining the degrees of freedom of the variances being tested. It has a probability distribution function :

• the triangular distribution, it gets the mode of the distribution as parameter. Dependent on the mode m the distribution is in the function takes on various triangular shapes. If the mode is equal to 0.5, the triangular distribution is symmetrical. It has the following probability function :