Counting process
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: N ( t ) ≥ 0. {\displaystyle N(t)\geq 0.} N ( t ) {\displaystyle N(t)} is an integer. 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.
A counting process is a stochastic process with values that are non-negative, integer, and non-decreasing:
- is an integer.
- If then
If , then is the number of events occurred during the interval Examples of counting processes include Poisson processes and Renewal processes.
![{\displaystyle (s,t].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1ec7ef02caed53f720f5996bd4d036dce7faa4)
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
[edit]- Intensity of counting processes
- Poisson point process (example for a counting process)
References
[edit]- 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