# Elementary flow

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 (*).

#### 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://www.encyclopediaofmath.org/index.php?title=Elementary_flow&oldid=26440