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

1. N ( t ) ≥ ≥ 0. {\displaystyle N(t)\geq 0.}

2. N ( t ) {\displaystyle N(t)} is an integer.

3. If s ≤ ≤ t {\displaystyle s\leq t} then N ( s ) ≤ ≤ N ( t ) . {\displaystyle N(s)\leq N(t).}

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

• References

• Intensity of counting processes

• Poisson point process (example for a counting process)

• 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