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{{MSC|60G55|60K25}}
 
{{MSC|60G55|60K25}}
  
 
[[Category:Stochastic processes]]
 
[[Category:Stochastic processes]]
  
A random sequence of moments of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353201.png" /> at which the events of a flow of events take place (e.g. a flow of incoming calls at a telephone station), and for which the differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353202.png" /> satisfy the condition of independence and have the same exponential distribution. An elementary flow with distribution
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A random sequence of moments of time $  0 < \tau _ {1} < \tau _ {2} < {} \dots $
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at which the events of a flow of events take place (e.g. a flow of incoming calls at a telephone station), and for which the differences $  \tau _ {i+} 1 - \tau _ {i} $
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satisfy the condition of independence and have the same exponential distribution. An elementary flow with distribution
  
<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/e/e035/e035320/e0353203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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F ( x)  = {\mathsf P}
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\{ \tau _ {i+} 1 - \tau _ {i} \leq  x \}
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= 1 - e ^ {- \lambda x } ,\ \
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x \geq  0 ,
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$$
  
is a particular case of a renewal process (cf. [[Renewal theory|Renewal theory]]). To an elementary flow is related the [[Poisson process|Poisson process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353204.png" /> equal to the number of events of the flow in the time interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353205.png" />. An elementary flow and its related Poisson process satisfy the following conditions.
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is a particular case of a renewal process (cf. [[Renewal theory|Renewal theory]]). To an elementary flow is related the [[Poisson process|Poisson process]] $  \xi ( t ) $
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equal to the number of events of the flow in the time interval $  ( 0 , t ) $.  
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An elementary flow and its related Poisson process satisfy the following conditions.
  
Stationarity. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353207.png" /> the distribution of the random variable
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Stationarity. For any $  0 < t _ {0} $,  
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$  0 < t _ {1} < \dots < t _ {k} $
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the distribution of the random variable
  
<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/e/e035/e035320/e0353208.png" /></td> </tr></table>
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$$
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\xi ( t _ {l} + t _ {0} ) - \xi
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( t _ {l-} 1 + t _ {0} ) ,\ \
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l = 2 \dots k ,
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$$
  
does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353209.png" />.
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does not depend on $  t _ {0} $.
  
Orderliness. The probability of occurrence of two or more events of the flow in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e03532010.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e03532011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e03532012.png" />.
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Orderliness. The probability of occurrence of two or more events of the flow in the interval $  ( t , t + \Delta t ) $
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is equal to $  o ( \Delta t ) $
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as $  t \rightarrow 0 $.
  
Lack of memory. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e03532013.png" /> the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e03532014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e03532015.png" />, are independent.
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Lack of memory. For $  0 < t _ {1} < \dots < t _ {n} $
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the random variables $  \xi ( t _ {l} ) - \xi ( t _ {l-} 1 ) $,  
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$  l = 1 \dots n $,  
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are independent.
  
 
It turns out that in these circumstances and under the condition
 
It turns out that in these circumstances and under the condition
  
<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/e/e035/e035320/e03532016.png" /></td> </tr></table>
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$$
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{\mathsf P} \{ \xi ( t + \Delta t ) - \xi ( t ) = 1 \}  = \
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\lambda \Delta t + o ( \Delta t )
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$$
  
 
the flow is elementary with exponential distribution (*).
 
the flow is elementary with exponential distribution (*).

Latest revision as of 19:37, 5 June 2020


2020 Mathematics Subject Classification: Primary: 60G55 Secondary: 60K25 [MSN][ZBL]

A random sequence of moments of time $ 0 < \tau _ {1} < \tau _ {2} < {} \dots $ at which the events of a flow of events take place (e.g. a flow of incoming calls at a telephone station), and for which the differences $ \tau _ {i+} 1 - \tau _ {i} $ satisfy the condition of independence and have the same exponential distribution. An elementary flow with distribution

$$ \tag{* } F ( x) = {\mathsf P} \{ \tau _ {i+} 1 - \tau _ {i} \leq x \} = 1 - e ^ {- \lambda x } ,\ \ x \geq 0 , $$

is a particular case of a renewal process (cf. Renewal theory). To an elementary flow is related the Poisson process $ \xi ( t ) $ equal to the number of events of the flow in the time interval $ ( 0 , t ) $. An elementary flow and its related Poisson process satisfy the following conditions.

Stationarity. For any $ 0 < t _ {0} $, $ 0 < t _ {1} < \dots < t _ {k} $ the distribution of the random variable

$$ \xi ( t _ {l} + t _ {0} ) - \xi ( t _ {l-} 1 + t _ {0} ) ,\ \ l = 2 \dots k , $$

does not depend on $ t _ {0} $.

Orderliness. The probability of occurrence of two or more events of the flow in the interval $ ( t , t + \Delta t ) $ is equal to $ o ( \Delta t ) $ as $ t \rightarrow 0 $.

Lack of memory. For $ 0 < t _ {1} < \dots < t _ {n} $ the random variables $ \xi ( t _ {l} ) - \xi ( t _ {l-} 1 ) $, $ l = 1 \dots n $, are independent.

It turns out that in these circumstances and under the condition

$$ {\mathsf P} \{ \xi ( t + \Delta t ) - \xi ( t ) = 1 \} = \ \lambda \Delta t + o ( \Delta t ) $$

the flow is elementary with exponential distribution (*).

References

[K] A.Ya. Khinchin, "Mathematical methods in the theory of queueing" , Griffin (1960) (Translated from Russian)
How to Cite This Entry:
Elementary flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elementary_flow&oldid=46802
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