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[[Category:Stochastic processes]]
 
[[Category:Stochastic processes]]
  
A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733401.png" /> with independent increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733403.png" />, having a [[Poisson distribution|Poisson distribution]]. In the homogeneous Poisson process
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
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{\mathsf P} \{ X ( t _ {2} ) - X ( t _ {1} ) = k \}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733405.png" /></td> </tr></table>
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\frac{\lambda  ^ {k} ( t _ {2} - t _ {1} )  ^ {k} }{k!}
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e ^ {- \lambda ( t _ {2} - t _ {1} ) } ,
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733406.png" />. The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733407.png" /> is called the intensity of the Poisson process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733408.png" />. The trajectories of the Poisson process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p0733409.png" /> are step-functions with jumps of height 1. The jump points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334010.png" /> form an [[Elementary flow|elementary flow]] describing the demand flow in many queueing systems. The distributions of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334011.png" /> are independent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334012.png" /> and have exponential density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334014.png" />.
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$$
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= 0 , 1 \dots
 +
$$
  
One of the properties of a Poisson process is that the conditional distribution of the jump points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334015.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334016.png" /> is the same as the distribution of the [[Variational series|variational series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334017.png" /> independent samples with uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334018.png" />. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334019.png" /> is the variational series described above, then as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334022.png" /> one obtains in the limit the distribution of the jumps of the Poisson process.
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for any  $  t _ {2} > t _ {1} $.
 +
The coefficient  $  \lambda > 0 $
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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 $.
  
In an inhomogeneous process the intensity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334023.png" /> depends on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334024.png" /> and the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334025.png" /> is defined by the formula
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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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073340/p07334026.png" /></td> </tr></table>
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In an inhomogeneous process the intensity  $  \lambda ( t) $
 +
depends on the time  $  t $
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and the distribution of  $  X ( t _ {2} ) - X ( t _ {1} ) $
 +
is defined by the formula
  
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 [[#References|[3]]].
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$$
 +
{\mathsf P} \{ X ( t _ {2} ) - X ( t _ {1} ) = k \} = \
  
====References====
+
\frac{\left [ \int\limits _ { t _ {1} } ^ { {t _ 2 } } \lambda ( ud u \right ^ {k} }{k!}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Borovkov,  "Wahrscheinlichkeitstheorie" , Birkhäuser  (1976(Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.I. Gikhman,  A.V. Skorokhod,  M.I. Yadrenko,  "Probability theory and mathematical statistics" , Kiev (1979)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Feller,  "An introduction to probability theory and its applications" , '''2''' , Wiley  (1971pp. Chapt. 1</TD></TR></table>
+
  e ^ {- \int\limits _ { t _ {1} } ^ { {t _ 2 } } \lambda ( ud 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}}.
  
 +
====References====
 +
{|
 +
|valign="top"|{{Ref|B}}|| A.A. Borovkov, "Wahrscheinlichkeitstheorie" , Birkhäuser (1976) (Translated from Russian) {{MR|0410818}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|GSY}}|| I.I. Gikhman, A.V. Skorokhod, M.I. Yadrenko, "Probability theory and mathematical statistics" , Kiev (1979) (In Russian) {{MR|2026607}} {{ZBL|0673.60001}}
 +
|-
 +
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''' , Wiley (1971) pp. Chapt. 1
 +
|}
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Cohen,   "The single server queue" , North-Holland (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.G. Székely,   "Paradoxes in probability theory and mathematical statistics" , Reidel (1986)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|C}}|| J.W. Cohen, "The single server queue" , North-Holland (1982) {{MR|0668697}} {{ZBL|0481.60003}}
 +
|-
 +
|valign="top"|{{Ref|S}}|| G.G. Székely, "Paradoxes in probability theory and mathematical statistics" , Reidel (1986) {{MR|0880020}} {{ZBL|0605.60002}}
 +
|}

Latest revision as of 08:06, 6 June 2020


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

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
How to Cite This Entry:
Poisson process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_process&oldid=23169
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