Statistical distributions provide a method of simulating the variations that occur in timing (and other numbers) in any process involving people or machines or anything in nature. SIMUL8 contains a range of wide Statistical Distributions. The purpose of this section is to introduce these distributions and outline some typical purposes.

There are two main categories of distribution: discrete and continuous.

When sampled a continuous distribution returns a non-integer value, whereas a discrete distribution will return an integer.

If you flip a coin or roll a dice there are a finite set of possible outcomes. These finite outcomes define a discrete distribution. Other examples could be the result of a test where the result is pass or fail, or the nature of a part when categorized by Part Number or Type.

A continuous distribution has an uncountable number of possible outcomes. The duration of a journey will follow a continuous distribution - each incidence of a journey will take a marginally different time - or the time taken to perform a manual operation will be continuous. In each of these cases the times may be very similar, but when measured to several decimal places then differences will be seen.

Distributions can be Bounded - meaning that they can have finite upper and/or lower limits.

SIMUL8 allows you to create your own distributions, however certain classical statistical distributions are provided:

**Exponential**

A continuous distribution bounded on the lower side. Classically used to represent the time between random occurrences, such as entities arriving into a system, where the arrivals are independent of each other and randomly distributed, and the occurrence of breakdowns.

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**Fixed**

This is not a distribution as such, but a static number that cannot vary. It will always produce the same result, unless altered through Visual Logic. A fixed value can be used for many purposes, such as timings that are fixed or setting label values.

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**Normal**

An unbounded continuous distribution, symmetrical in shape. When plotted it is generally referred to as a Bell Curve. The Normal distribution is generally used to generate processing or job times, however this may be incorrect. Processing times are generally more likely to be represented as a skewed distribution, such as Log normal. Additionally, it might also be used when label values are set for which it is very unlikely to obtain an extreme low or high value.

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**Uniform**

A continuous distribution bounded at the upper and lower limits. It is useful for situations where there is a random occurrence between the upper and lower values, but where little else is known, so that any sampled value is equally likely.

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**Average**

An unbounded continuous distribution with the same features of the Normal distribution and for which the standard deviation is pre-set to ±25%. It has the same common usages as the Normal distribution, however under consideration of the pre-set, fairly broad variability.

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**Beta**

A bounded continuous distribution useful for situations where data is sparse.

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**Bernoulli**

A discrete distribution with two possible outcomes, 0 or 1. Similar to flipping a coin, however unlike a coin there is not an equal probability of each value being sampled. It is commonly used for setting labels when they can take either of two values with certain probabilities. For example, work items passing a test with “Yes” or “No” and this is stored in a label for further use in the simulation, such as Routing Out.

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**Binomial**

A discrete distribution for the number of successes in a specified number of trials of (independent) binary experiments (0 or 1). It is related to the Bernoulli distribution and a common use is for setting label values, for example how often a work item went through a specified number of tests with a certain outcome. Other uses can be for the number of items in a batch, or the number of items demanded from inventory.

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**Erlang**

A continuous distribution bounded on the lower side. Useful in reliability modeling to simulate repair times.

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**Gamma**

A continuous distribution bounded at the lower side. This has been used to generate lifetimes, lead times, personal income data, a population about a stable equilibrium, inter arrival times, and service times.

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**Geometric**

A discrete distribution bounded at 0 on the lower side. It returns the number of failures before a success where there is a sequence of independent trials of binary experiments with outcomes 0 or 1. It is related to the Bernoulli distribution. An exemplary use in SIMUL8 is for setting a label value, such as the number of re-work loops that a work item needs based on the number of previously failed tests.

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**Log Normal**

A continuous distribution bounded on the lower side. It is commonly used to generate processing times.

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**Negative Binomial**

A discrete distribution bounded on the lower side at 0. The Negative Binomial distribution gives the total number of trials, x, to get k events (failures…), each with the constant probability, p, of occurring. It is commonly used to predict the number of failures in a batch or number of items demanded from inventory.

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**Pearson V**

A bounded continuous distribution, bounded at the lower side at a set parameter. It is useful for modeling time delays or process times where a minimum value is assumed.

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**Pearson VI**

A bounded continuous distribution, bounded on the lower side. It is generally used to generate the time taken for a task.

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**Poisson**

A discrete distribution bounded at 0 on the low side and unbounded on the high side. It is frequently used to simulate the occurrence of infrequent events whose rate is constant. This includes the occurrence of events such as telephone calls or defects in manufacturing.

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**Rounded Uniform**

A discrete distribution bounded at the lower and upper sides. There is an equal probability of any value between the lower and upper bounds being sampled. This is generally only used where there is an absence of quality data that can be used to identify the shape of the distribution, and is used for generating batch sizes or item types.

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**Triangular**

A continuous distribution, bounded at the upper and lower sides. Generally used only when data is sparse and where the distribution is discernibly not Uniform. Triangular uses 3 data points, Upper, Lower and Modal, to connect 3 points in a triangle.

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**Weibull**

A continuous distribution bounded at the lower side. It is a very flexible distribution used for many purposes. It is used to describe the duration of a task, lifetime of a piece of equipment and many other situations.

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