Poisson process

2020 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