# Statistics for Queueing Theory

Queueing Theory relies on a lot of statistical theory for it's underlying mathematics. This page contains some links to introductory statistics information written by third parties, and an exploration of some statistical distributions used in queueing theory. If the reader feels fully at home with the statistics explored in this page, it can be skipped. Also, the reader may find it easier in some cases to come back to the distributions in this page when they are explored in more detail. Appropriate links will be provided in these situations.

## Third-Party Introductions to Basic Statistics

Note: as these are links to external sites, their quality or reliability cannot be guaranteed. However, they were checked links as of the date this page was last updated, and were working reliably then.

## Statistical Distributions Used in Queueing Theory

While there are other distributions apart from those outlined below which could be used for queueing theory analysis purposes, this author has not encountered them, and these distributions seem by far to be the most popular, and are also, according to many sources, the distributions most tractable to mathematical analysis. More information on the distributions can be found in the third-party pages listed above.

### Degenerate (or Deterministic) Distribution

The variate in this distribution is always a constant value. This could be used, for example, to model the input of parts to a manufacturing machine, if one knew that the input occurred at a constant rate.

### Exponential and Erlang Distributions

The exponential distribution used in queueing theory is a negative exponential distribution. It's PDF (probability density function) is:

As can be seen from the equation above, the exponential distribution has one parameter which describes it - denoted in the equation by the letter a. The exponential distribution is a special case of the Erlang distribution (named after A. K. Erlang, one of the fathers of queueing theory). The PDF of the Erlang distribution is:

### General Distribution

This does not refer to any particular statistical distribution, rather, an arbitrary one. In other words, every other distribution is only a special case of the general distribution, and all results derived from use of the general distribution can be applied to any other distribution, with the addition of the appropriate restrictions. For queueing theory purposes, the only information one requires when applying results derived from the General Distribution, is the mean and variance of the distribution.

Last Updated: 14th June 1999
Written by: Andrew Ferrier