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.

Counting process

See also