# Poisson process

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2010 Mathematics Subject Classification: Primary: 60G55 Secondary: 60K25 [MSN][ZBL]

A stochastic process $X ( t)$ with independent increments $X ( t _ {2} ) - X ( t _ {1} )$, $t _ {2} > t _ {1}$, having a Poisson distribution. In the homogeneous Poisson process

$$\tag{1 } {\mathsf P} \{ X ( t _ {2} ) - X ( t _ {1} ) = k \} = \ \frac{\lambda ^ {k} ( t _ {2} - t _ {1} ) ^ {k} }{k!} e ^ {- \lambda ( t _ {2} - t _ {1} ) } ,$$

$$k = 0 , 1 \dots$$

for any $t _ {2} > t _ {1}$. The coefficient $\lambda > 0$ is called the intensity of the Poisson process $X ( t)$. The trajectories of the Poisson process $X ( t)$ are step-functions with jumps of height 1. The jump points $0 < \tau _ {1} < \tau _ {2} < \dots$ form an elementary flow describing the demand flow in many queueing systems. The distributions of the random variables $\tau _ {n} - \tau _ {n-} 1$ are independent for $n = 1 , 2 \dots$ and have exponential density $\lambda e ^ {- \lambda t }$, $t \geq 0$.

One of the properties of a Poisson process is that the conditional distribution of the jump points $0 < \tau _ {1} < \dots < \tau _ {n} < t$ when $X ( t) - X ( 0) = n$ is the same as the distribution of the variational series of $n$ independent samples with uniform distribution on $[ 0 , t ]$. On the other hand, if $0 < \tau _ {1} < \dots < \tau _ {n}$ is the variational series described above, then as $n \rightarrow \infty$, $t \rightarrow \infty$ and $n / t \rightarrow \lambda$ one obtains in the limit the distribution of the jumps of the Poisson process.

In an inhomogeneous process the intensity $\lambda ( t)$ depends on the time $t$ and the distribution of $X ( t _ {2} ) - X ( t _ {1} )$ is defined by the formula

$${\mathsf P} \{ X ( t _ {2} ) - X ( t _ {1} ) = k \} = \ \frac{\left [ \int\limits _ { t _ {1} } ^ { {t _ 2 } } \lambda ( u) d u \right ] ^ {k} }{k!} e ^ {- \int\limits _ { t _ {1} } ^ { {t _ 2 } } \lambda ( u) d u } .$$

Under certain conditions a Poisson process can be shown to be the limit of the sum of a number of independent "sparse" flows of fairly general form as this number increases to infinity. For certain paradoxes which have been obtained in connection with Poisson processes see [F].

#### References

 [B] A.A. Borovkov, "Wahrscheinlichkeitstheorie" , Birkhäuser (1976) (Translated from Russian) MR0410818 [GSY] I.I. Gikhman, A.V. Skorokhod, M.I. Yadrenko, "Probability theory and mathematical statistics" , Kiev (1979) (In Russian) MR2026607 Zbl 0673.60001 [F] W. Feller, "An introduction to probability theory and its applications", 2 , Wiley (1971) pp. Chapt. 1

#### References

 [C] J.W. Cohen, "The single server queue" , North-Holland (1982) MR0668697 Zbl 0481.60003 [S] G.G. Székely, "Paradoxes in probability theory and mathematical statistics" , Reidel (1986) MR0880020 Zbl 0605.60002
How to Cite This Entry:
Poisson process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_process&oldid=48220
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article