Difference between revisions of "Poisson process"
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{{MSC|60G55|60K25}} | {{MSC|60G55|60K25}} | ||
[[Category:Stochastic processes]] | [[Category:Stochastic processes]] | ||
| − | A [[Stochastic process|stochastic process]] | + | A [[Stochastic process|stochastic process]] $ X ( t) $ |
| + | with independent increments $ X ( t _ {2} ) - X ( t _ {1} ) $, | ||
| + | $ t _ {2} > t _ {1} $, | ||
| + | having a [[Poisson distribution|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|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|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 {{Cite|F}}. | 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 {{Cite|F}}. | ||
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====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
Latest revision as of 08:06, 6 June 2020
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 |
Comments
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 |
Poisson process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_process&oldid=48220