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19.4 Unsteady Poisson Flow

Lecture



If the flow of events is non-stationary, then its main characteristic is instantaneous density.   19.4 Unsteady Poisson Flow . The instantaneous flux density is the limit of the ratio of the average number of events per elementary time interval.   19.4 Unsteady Poisson Flow , to the length of this section, when the latter tends to zero:

  19.4 Unsteady Poisson Flow , (19.4.1)

Where   19.4 Unsteady Poisson Flow - mathematical expectation of the number of events at the site   19.4 Unsteady Poisson Flow .

Consider the flow of homogeneous events, ordinary and without aftereffect, but not stationary, with variable density   19.4 Unsteady Poisson Flow . Such a stream is called a non-stationary Poisson stream. This is the first step of generalization compared to the simplest flow. It is easy to show a method similar to that used in   19.4 Unsteady Poisson Flow 5.9, that for such a stream the number of events falling on the length segment   19.4 Unsteady Poisson Flow starting at point   19.4 Unsteady Poisson Flow obeys the Poisson law

  19.4 Unsteady Poisson Flow   19.4 Unsteady Poisson Flow , (19.4.2)

Where   19.4 Unsteady Poisson Flow - mathematical expectation of the number of events in the area from   19.4 Unsteady Poisson Flow before   19.4 Unsteady Poisson Flow equal to

  19.4 Unsteady Poisson Flow . (19.4.3)

Here is the value   19.4 Unsteady Poisson Flow depends not only on length   19.4 Unsteady Poisson Flow plot, but also from its position on the axis   19.4 Unsteady Poisson Flow .

We find for a nonstationary flow the distribution law of the time interval   19.4 Unsteady Poisson Flow between neighboring events. Due to the nonstationarity of the flow, this law will depend on where on the axis   19.4 Unsteady Poisson Flow The first event is located. In addition, it will depend on the type of function   19.4 Unsteady Poisson Flow . Suppose that the first of two adjacent events appeared at the moment   19.4 Unsteady Poisson Flow and find under this condition the law of time distribution   19.4 Unsteady Poisson Flow between this event and the following:

  19.4 Unsteady Poisson Flow .

We find   19.4 Unsteady Poisson Flow - the probability that in the area from   19.4 Unsteady Poisson Flow before   19.4 Unsteady Poisson Flow no events will appear:

  19.4 Unsteady Poisson Flow ,

from where

  19.4 Unsteady Poisson Flow . (19.4.4)

Differentiating, we find the distribution density

  19.4 Unsteady Poisson Flow   19.4 Unsteady Poisson Flow . (19.4.5)

This distribution law is no longer indicative. Its appearance depends on the parameter.   19.4 Unsteady Poisson Flow and kind of function   19.4 Unsteady Poisson Flow . For example, with a linear change   19.4 Unsteady Poisson Flow

  19.4 Unsteady Poisson Flow

density (19.4.5) is

  19.4 Unsteady Poisson Flow . (19.4.6)

The schedule of this law is   19.4 Unsteady Poisson Flow ;   19.4 Unsteady Poisson Flow and   19.4 Unsteady Poisson Flow presented in fig. 19.4.1.

  19.4 Unsteady Poisson Flow

Fig. 19.4.1.

Despite the fact that the structure of a nonstationary Poisson flow is somewhat more complicated than the simplest one, it is very convenient in practical applications: the main property of the simplest flow — the absence of aftereffect — is preserved in it. Namely, if we fix on the axis   19.4 Unsteady Poisson Flow arbitrary point   19.4 Unsteady Poisson Flow then the distribution law   19.4 Unsteady Poisson Flow of time   19.4 Unsteady Poisson Flow , separating this point from the nearest future event, does not depend on what happened at the time interval preceding   19.4 Unsteady Poisson Flow and at the very point   19.4 Unsteady Poisson Flow (i.e., whether other events have appeared earlier and when).


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Queuing theory

Terms: Queuing theory