View source for Elementary flow

{{MSC|60G55|60K25}}

[[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

<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>

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.

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

<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>

does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035320/e0353209.png" />.

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" />.

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.

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>

the flow is elementary with exponential distribution (*).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,   "Mathematical methods in the theory of queueing" , Griffin  (1960)  (Translated from Russian)</TD></TR></table>