Counting process

Example of a counting process: the Poisson process, here with different intensities

A counting process is a special stochastic process . It counts the number of events of a certain type that have occurred up to a point in time.

Application and examples

Counting processes play a role in the modeling and simulation of operating systems , but the simulation is usually only practicable for special counting processes, especially renewal processes through event-oriented simulation .

The Poisson process , which is also a renewal process, can be mentioned as an example of a counting process . Some theories can be represented using counting processes. For example, the entire survival time  analysis can be represented using the counting process theory.

definition

A stochastic process is called a counting process if there is a sequence of positive random variables , the so-called inter- arrival times, such that : ${\displaystyle N=(N(t))_{t\geq 0}}$ ${\displaystyle (Y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle S_{n}:=Y_{1}+\dotsb +Y_{n}}$

${\displaystyle N(t)=\left\{{\begin{matrix}0,&{\mbox{falls }}t<Y_{1}\\\max \left\{n\in \mathbb {N} :S_{n}\leq t\right\},&{\mbox{sonst}}\end{matrix}}\right.}$

The result of the then forms a point process on . This positions random points on the positive number line, the distances between which are distributed according to the inter-arrival times. The counting process then runs through the positive numbers at a constant speed and counts how many points it has already hit up to the point in time . Because of this close connection, the counting process and the point process are sometimes used synonymously. ${\displaystyle S_{n}}$ ${\displaystyle \mathbb {R} ^{+}}$ ${\displaystyle t}$

The paths of a counting process are therefore jump functions with jump height 1.

Individual evidence

1. ↑ Jochen Wengenroth: Probability Theory. de Gruyter, Berlin 2008, ISBN 978-3-11-020358-5 , p. 127 ( limited preview in the Google book search).
2. ↑ Eric W. Weisstein : Point Process . In: MathWorld (English).