Elementary flow

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2010 Mathematics Subject Classification: Primary: 60G55 Secondary: 60K25 [MSN][ZBL]

A random sequence of moments of time 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 satisfy the condition of independence and have the same exponential distribution. An elementary flow with distribution


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

Stationarity. For any , the distribution of the random variable

does not depend on .

Orderliness. The probability of occurrence of two or more events of the flow in the interval is equal to as .

Lack of memory. For the random variables , , are independent.

It turns out that in these circumstances and under the condition

the flow is elementary with exponential distribution (*).


[K] A.Ya. Khinchin, "Mathematical methods in the theory of queueing" , Griffin (1960) (Translated from Russian)
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Elementary flow. Encyclopedia of Mathematics. URL:
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