# 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 {if}} t

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).