Counting process

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

1. $N(t)\ge 0.$
2. $N(t)$ is an integer.
3. If $s\le t$ then $N(s)\le N(t).$

If $s<t$, 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.

See also

• Intensity of counting processes
• Poisson point process (example for a 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.