# Elementary flow

2010 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