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.