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19.5. Stream with limited aftereffect (stream Palma)

Lecture



In the previous   19.5.  Stream with limited aftereffect (stream Palma) we got acquainted with the natural generalization of the simplest flow — the unsteady Poisson flow. A generalization of the simplest flow in the other direction is a flow with limited aftereffect.

Consider the ordinary stream of homogeneous events (Fig. 19.5.1).

  19.5.  Stream with limited aftereffect (stream Palma)

Fig. 19.5.1.

This stream is called a stream with limited aftereffect (or a Palma stream) if the intervals between successive events   19.5.  Stream with limited aftereffect (stream Palma) are independent random variables.

Obviously, the simplest flow is a special case of the flow of Palma: there are distances in it.   19.5.  Stream with limited aftereffect (stream Palma) are independent random variables distributed according to exponential law. As for the unsteady Poisson flow, it is not a Palma flow. Indeed, consider the two adjacent intervals   19.5.  Stream with limited aftereffect (stream Palma) and   19.5.  Stream with limited aftereffect (stream Palma) in non-stationary Poisson flow. As we saw in the previous   19.5.  Stream with limited aftereffect (stream Palma) , the law of distribution of the gap between events in a non-stationary stream depends on where this gap begins, and the beginning of the gap   19.5.  Stream with limited aftereffect (stream Palma) coincides with the end of the gap   19.5.  Stream with limited aftereffect (stream Palma) ; This means that the lengths of these intervals are dependent.

Consider the examples of Palma flows.

1. Some part of a technical device (for example, a radio tube) works continuously until its failure (failure), after which it is instantly replaced by a new one. The term of no-failure operation is accidental; some instances fail independently of each other. Under these conditions, the flow of failures (or the flow of "recovery") is the flow of Palm. If, moreover, the period of work of the part is distributed according to the exponential law, then the Palm flow turns into the simplest one.

2. A group of aircraft goes into battle order "column" (Fig. 19.5.2) with the same speed for all aircraft   19.5.  Stream with limited aftereffect (stream Palma) . Each aircraft, except the lead, is obliged to maintain operation, i.e. to keep at a given distance   19.5.  Stream with limited aftereffect (stream Palma) from going ahead. This distance, due to the errors of the radio range finder, is maintained with errors. The moments of the intersection of a given line of aircraft form a stream of Palma, since random variables   19.5.  Stream with limited aftereffect (stream Palma) ;   19.5.  Stream with limited aftereffect (stream Palma) ; ... are independent. Note that the same flow will not be the flow of Palma, if each of the aircraft tends to maintain a given distance not from the neighbor, but from the leader.

  19.5.  Stream with limited aftereffect (stream Palma)

Fig. 19.5.2.

Palma flows are often obtained as output flows of queuing systems. If any system receives a stream of requests, then it is divided into two by this system: the stream of serviced and the stream of unserved requests.

The flow of unserved applications is often sent to some other queuing system, so it is of interest to study its properties.

The main one in the theory of output flows is the Palma theorem, which we state without proof.

Let the flow of requests like Palma arrive at the queuing system, and an application that has made all channels busy gets rejected (not served). If, at the same time, the service time has an exponential distribution law, then the flow of unserved applications is also a Palm type flow.

In particular, if the input flow of applications is the simplest, then the stream of unserved applications, without being the simplest, will still have a limited follow-up.

An interesting example of streams with limited aftereffect is the so-called Erlang streams. They are formed by "screening" the simplest flow.

Consider the simplest flow (Fig. 19.5.3) and discard every second point from it (in the figure, the ejected points are marked with crosses).

  19.5.  Stream with limited aftereffect (stream Palma)

Fig. 19.5.3.

The remaining points form a stream; this flow is called the first order Erlang flow   19.5.  Stream with limited aftereffect (stream Palma) . Obviously, this flow is the Palm flow: since the intervals between events in the simplest flow are independent, the values   19.5.  Stream with limited aftereffect (stream Palma) , obtained by summing up such intervals in two.

The second-order Erlang flow will be obtained if every third point is kept in the simplest flow, and two intermediate points are discarded (Fig. 19.5.4).

  19.5.  Stream with limited aftereffect (stream Palma)

Fig. 19.5.4.

In general, the k-th order Erlang flow   19.5.  Stream with limited aftereffect (stream Palma) called a stream derived from the simplest if you save each   19.5.  Stream with limited aftereffect (stream Palma) point and throw the rest. Obviously, the simplest flow can be considered as a zero order Erlang flow.   19.5.  Stream with limited aftereffect (stream Palma) .

Find the distribution law of the time interval   19.5.  Stream with limited aftereffect (stream Palma) between neighboring events in the Erlang stream   19.5.  Stream with limited aftereffect (stream Palma) th order   19.5.  Stream with limited aftereffect (stream Palma) . Consider on the axis   19.5.  Stream with limited aftereffect (stream Palma) (fig. 19.5.5) the simplest flow with intervals   19.5.  Stream with limited aftereffect (stream Palma)

  19.5.  Stream with limited aftereffect (stream Palma)

Fig. 19.5.5.

Magnitude   19.5.  Stream with limited aftereffect (stream Palma) represents the sum   19.5.  Stream with limited aftereffect (stream Palma) independent random variables

  19.5.  Stream with limited aftereffect (stream Palma) , (19.5.1)

Where   19.5.  Stream with limited aftereffect (stream Palma) - independent random variables subject to the same indicative law

  19.5.  Stream with limited aftereffect (stream Palma)   19.5.  Stream with limited aftereffect (stream Palma) . (19.5.2)

It would be possible to find the distribution law   19.5.  Stream with limited aftereffect (stream Palma) as a composition   19.5.  Stream with limited aftereffect (stream Palma) laws (19.5.2). However, it is easier to derive it by elementary reasoning.

Denote   19.5.  Stream with limited aftereffect (stream Palma) distribution density   19.5.  Stream with limited aftereffect (stream Palma) for flow   19.5.  Stream with limited aftereffect (stream Palma) ;   19.5.  Stream with limited aftereffect (stream Palma) there is a probability that the magnitude   19.5.  Stream with limited aftereffect (stream Palma) will take the value between   19.5.  Stream with limited aftereffect (stream Palma) and   19.5.  Stream with limited aftereffect (stream Palma) (fig. 19.5.5). This means that the last point of the gap   19.5.  Stream with limited aftereffect (stream Palma) should get to the elementary site   19.5.  Stream with limited aftereffect (stream Palma) and previous   19.5.  Stream with limited aftereffect (stream Palma) points of the simplest flow - on the plot   19.5.  Stream with limited aftereffect (stream Palma) . The probability of the first event is equal to   19.5.  Stream with limited aftereffect (stream Palma) the probability of the second, based on the formula (19.3.2), will be

  19.5.  Stream with limited aftereffect (stream Palma) .

Multiplying these probabilities, we get

  19.5.  Stream with limited aftereffect (stream Palma) ,

from where

  19.5.  Stream with limited aftereffect (stream Palma)   19.5.  Stream with limited aftereffect (stream Palma) . (19.5.3)

The distribution law with density (19.5.3) is called the Erlang law   19.5.  Stream with limited aftereffect (stream Palma) th order. Obviously when   19.5.  Stream with limited aftereffect (stream Palma) he turns into indicative

  19.5.  Stream with limited aftereffect (stream Palma)   19.5.  Stream with limited aftereffect (stream Palma) . (19.5.4)

Find the characteristics of the Erlang law   19.5.  Stream with limited aftereffect (stream Palma) : expected value   19.5.  Stream with limited aftereffect (stream Palma) and variance   19.5.  Stream with limited aftereffect (stream Palma) . By the theorem of addition of mathematical expectations

  19.5.  Stream with limited aftereffect (stream Palma) ,

Where   19.5.  Stream with limited aftereffect (stream Palma) - expectation of the gap between events in the simplest flow.

From here

  19.5.  Stream with limited aftereffect (stream Palma) . (19.5.5)

Similarly, by the addition theorem

  19.5.  Stream with limited aftereffect (stream Palma) ,   19.5.  Stream with limited aftereffect (stream Palma) . (19.5.6)

Density   19.5.  Stream with limited aftereffect (stream Palma) flow   19.5.  Stream with limited aftereffect (stream Palma) will be the opposite of   19.5.  Stream with limited aftereffect (stream Palma)

  19.5.  Stream with limited aftereffect (stream Palma) . (19.5.7)

Thus, with an increase in the order of the Erlang flow, both the expectation and the variance of the time interval between events increase, and the density of the flow decreases.

Find out how the Erlang flow will change when   19.5.  Stream with limited aftereffect (stream Palma) if its density will be kept constant? We normalize   19.5.  Stream with limited aftereffect (stream Palma) so that its expectation (and, therefore, flux density) remains unchanged. To do this, change the scale along the time axis and instead of   19.5.  Stream with limited aftereffect (stream Palma) consider the value

  19.5.  Stream with limited aftereffect (stream Palma) . (19.5.8)

Let's call such a stream the normal Erlang flow   19.5.  Stream with limited aftereffect (stream Palma) th order. Gap distribution law   19.5.  Stream with limited aftereffect (stream Palma) between the events of this thread will be

  19.5.  Stream with limited aftereffect (stream Palma)   19.5.  Stream with limited aftereffect (stream Palma) , (19.5.9)

Where   19.5.  Stream with limited aftereffect (stream Palma) , or

  19.5.  Stream with limited aftereffect (stream Palma)   19.5.  Stream with limited aftereffect (stream Palma) . (19.5.10)

Expectation value   19.5.  Stream with limited aftereffect (stream Palma) distributed according to the law (19.5.10) does not depend on   19.5.  Stream with limited aftereffect (stream Palma) and equals

  19.5.  Stream with limited aftereffect (stream Palma) ,

Where   19.5.  Stream with limited aftereffect (stream Palma) - the flux density coinciding with any   19.5.  Stream with limited aftereffect (stream Palma) with the density of the original simplest flow. Variance of magnitude   19.5.  Stream with limited aftereffect (stream Palma) equals

  19.5.  Stream with limited aftereffect (stream Palma) (19.5.11)

and decreases indefinitely with increasing   19.5.  Stream with limited aftereffect (stream Palma) .

Thus, we conclude: with an unlimited increase   19.5.  Stream with limited aftereffect (stream Palma) the normal Erlang flow approaches a regular flow with constant intervals equal to   19.5.  Stream with limited aftereffect (stream Palma) .

This property of the Erlang flows is convenient in practical applications: it gives the possibility, by asking various   19.5.  Stream with limited aftereffect (stream Palma) , get any degree of aftereffect: from total absence   19.5.  Stream with limited aftereffect (stream Palma) up to a hard functional connection between the events   19.5.  Stream with limited aftereffect (stream Palma) . Thus, the order of the flow of the Erlang can serve as a "measure of the aftereffect" existing in the flow. In order to simplify, it is often convenient to replace the real flow of applications, which has an after-effect, with the normalized Erlang flow with approximately the same characteristics of the gap between applications: expected value and variance.

Example. As a result of statistical processing of the intervals between applications in the stream, estimates are obtained for the expectation and variance of   19.5.  Stream with limited aftereffect (stream Palma) :

  19.5.  Stream with limited aftereffect (stream Palma) (min)   19.5.  Stream with limited aftereffect (stream Palma) (min2)

Replace this stream with a normal Erlang stream with the same characteristics.

Decision. We have

  19.5.  Stream with limited aftereffect (stream Palma) .

From the formula (19.5.11) we get

  19.5.  Stream with limited aftereffect (stream Palma) ,   19.5.  Stream with limited aftereffect (stream Palma) .

The flow can be approximately replaced by a fourth-order normalized Erlang flow.

created: 2017-07-03
updated: 2021-03-13
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Queuing theory

Terms: Queuing theory