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{{MSC|60G55|60K25}}

[[Category:Stochastic processes]]

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}}.

====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====

====References====
{|
|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}}
|}