Counting process

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title: "Counting process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["stochastic-processes"] topic_path: "general/stochastic-processes" source: "https://en.wikipedia.org/wiki/Counting_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

A counting process is a stochastic process {N(t), t\geq0} with values that are non-negative, integer, and non-decreasing:

  1. N(t)\geq0.
  2. N(t) is an integer.
  3. If s\leq t then N(s)\leq N(t).

If s, then N(t)-N(s) is the number of events occurred during the interval (s,t]. Examples of counting processes include Poisson processes and Renewal processes.

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, it is said to be a Markov counting process.

References

  • Ross, S.M. (1995) Stochastic Processes. Wiley.
  • Higgins JJ, Keller-McNulty S (1995) Concepts in Probability and Stochastic Modeling. Wadsworth Publishing Company.

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stochastic-processes