Elementary flow

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)