# A Petri Net Approach for Business Process Modeling and Simulation
## Appendix A Probability Distributions
For a sample of random numbers to follow a gamma distribution, it must have the following characteristics:  
* Form of the gamma distribution: the gamma distribution is a continuous probability distribution that is generally used to model the time between events in Poisson processes or to describe wait times. It has a shape that can be asymmetric and that varies depending on the parameters of the distribution. * Parameters of the gamma distribution: the gamma distribution is defined by two parameters, namely, the shape parameter (\(\kappa\)) and the scale parameter (\(\theta\)). These parameters control the shape and scale of the distribution. The shape parameter determines the shape of the curve (higher values of \(\kappa\) produce a more skewed distribution to the right), while the scale parameter adjusts the scale of the values. * Definite probability density function: the probability density function (PDF) of the gamma distribution is defined as a function of the parameters \(\kappa\) and \(\theta\), and is a mathematical function that describes the probability that a random variable takes a value in a specified interval. * Positive values: the gamma distribution is defined only for positive values. This is due to the nature of the application, as it is typically used to model event times or rates. * Independence: the values in a sample that follow a gamma distribution are independent of each other. This means that the occurrence of one value does not affect the occurrence of other values. * Applications in waiting processes: the gamma distribution is useful for modeling the time between events in situations where events occur in a Poisson process, such as time between phone calls, queue arrivals, or equipment failures, among others. **Exponential Distribution. ** This distribution models the time between events in a Poisson process, such as the time between failures in a system. For a sample of random numbers to follow an exponential distribution, it must have the following characteristics:  
* Waiting process model: the exponential distribution is commonly used to model the time between events in a Poisson process. Therefore, it is especially applicable for modeling events that occur independently at a constant rate. * Exponential decay: the probability density function (PDF) of the exponential distribution has an exponential decay form. This means that the probability of an event occurring in a given interval decreases exponentially as time passes. * Rate parameter: the exponential distribution is defined by a single parameter, often denoted as \(\lambda\) (lambda), which represents the average rate of occurrence of events. Its inverse value is known as the scale parameter (\(\beta=1/\lambda\)). * Positive values: as with other time-related distributions, the exponential distribution is defined only for positive values, as it is designed to model wait times between events. * Independence: the times between events in an exponential distribution are independent. The occurrence of one event does not affect the occurrence of future events. * Lack of memory: lack of memory means that the probability of an event occurring in a future interval does not depend on how much time has elapsed since the last event. In other words, the exponential distribution has no \"memory\" of past events.