# Counting process

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Counting_process
> Markdown URL: https://mediated.wiki/source/Counting_process.md
> Source: https://en.wikipedia.org/wiki/Counting_process
> Source revision: 1289721052
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

A '''counting process''' is a [stochastic process](/source/stochastic_process) <math>\{N(t), t\geq0\}</math> with values that are non-negative, integer, and non-decreasing:

# <math>N(t)\geq0.</math>
# <math>N(t)</math> is an integer.
# If <math>s\leq t</math> then <math>N(s)\leq N(t).</math>

If <math>s<t</math>, then <math>N(t)-N(s)</math> is the number of events occurred during the interval <math>(s,t].</math> Examples of counting processes include [Poisson process](/source/Poisson_process)es and [Renewal processes](/source/Renewal_theory).

Counting processes deal with the number of occurrences of something over time. An example of a counting process is the number of job arrivals to a queue over time.

If a process has the [Markov property](/source/Markov_property), it is said to be a Markov counting process.

==See also==
*[Intensity of counting processes](/source/Intensity_of_counting_processes)
*[Poisson point process](/source/Poisson_point_process) (example for a counting process)

==References==

* Ross, S.M. (1995) ''Stochastic Processes''. Wiley. {{ISBN|978-0-471-12062-9}}
* Higgins JJ, Keller-McNulty S (1995) ''Concepts in Probability and Stochastic Modeling''. Wadsworth Publishing Company. {{ISBN|0-534-23136-5}}

Category:Stochastic processes

---
Adapted from the Wikipedia article [Counting process](https://en.wikipedia.org/wiki/Counting_process) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Counting_process?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.