Counting process

A counting process is a stochastic process ${\displaystyle \{N(t),t\geq 0\}}$ with values that are non-negative, integer, and non-decreasing:

1. ${\displaystyle N(t)\geq 0.}$
2. ${\displaystyle N(t)}$ is an integer.
3. If ${\displaystyle s\leq t}$ then ${\displaystyle N(s)\leq N(t).}$

If ${\displaystyle s<t}$, then ${\displaystyle N(t)-N(s)}$ is the number of events occurred during the interval ${\displaystyle (s,t].}$ 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.

See also

• Intensity of counting processes
• 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