Poisson process. Stationary Poisson flow of failures Machine failures form a Poisson flow with an intensity

Computer science, cybernetics and programming

Definition of Poisson flow. Poisson flow is an ordinary flow without aftereffect. The classic traffic model in information networks is the Poisson simplest flow. It is characterized by a set of probabilities Pk of the arrival of k messages during the time interval t: where k \u003d 01 is the number of messages; λ flow rate.

1. Definition of Poisson flow. Properties.

Poisson flow is an ordinary flow without aftereffect.

The classical model of traffic in information networks is the Poisson (simplest) flow. It is characterized by a set of probabilities P (k) of the arrival of k messages over the time interval t:

where k \u003d 0.1, ... is the number of messages; λ - flow rate.

Note that the time interval for measuring the number of messages t and the flow rate λ are constant values.

The family of Poisson distributions P (k) depending on λ is shown in Fig. 1. A larger λ value corresponds to a wider and more symmetrical probability density plot.

Figure: 1. Poisson distributions. Probability densities.

The expectation (mean) and variance of the Poisson flow are equal to λt.

Knowing the probability of data arrival for a period, one can obtain the distribution of the interval τ between neighboring events:

Hence the conclusion: Poisson flow is characterized by an exponential distribution of intervals between events.

The main property of Poisson flow, which determines its widespread use in modeling, is additivity: the resulting flow of the sum of Poisson flows is also Poisson with the total intensity:

In modeling, the Poisson stream can be obtained by multiplexing a set of ON / OFF sources, which are called Markov processes (Fig. 2).

Figure: 2. Obtaining the Poisson distribution

2. QS with failures (classical Erlang system)

Here we consider one of the first in time, "classical" problems of the theory queuing; this problem arose from the practical needs of telephony and was solved in 1909 by the Danish engineer-mathematician A.K. Erlang. The problem is posed as follows: there are n channels (communication lines), which receive a flow of requests with intensity λ. The service flow of each channel has intensity μ. Find the limiting probabilities of system states and indicators of its efficiency.

System S (QS) has the following states (we number them according to the number of requests in the system): S0, S 1, ..., S n, where S k - the state of the system when there are k customers in it, i.e. k channels are occupied.

The QS state graph corresponds to the process of death and reproduction (Fig. 3).

Figure: 3. State graph of the QS

The flow of claims sequentially transfers the system from any left state to an adjacent right one with the same intensity λ. The intensity of the flow of services that transfer the system from any right state to an adjacent left one constantly changes depending on the state. Indeed, if the QS is in the state S2 (two channels are busy), then it can go to state S1 (one channel is busy) when either the first or the second channel finishes serving, i.e. the total intensity of their service flows will be 2μ. Similarly, the total flow of services transferring the QS from state S3 (three channels are busy) in S2 , will have an intensity of 3μ, i.e. any of the three channels can be released, etc.

In formula (1) for the scheme of death and reproduction, we obtain for the limiting probability of the state:

(1)

where the expansion terms are the coefficients at p0 in expressions for limiting probabilities p1, p 2, ..., p n.

Note that the intensities λ and μ enter into formula (1) not separately, but only in the form of the ratio μ / λ. We denote: μ / λ \u003dp , and we will call the value ρ the reduced rate of the flow of claims or the rate of the channel load. It expresses the average number of customers arriving during the average service time of one customer. Using this notation, we rewrite formula (1) in the form:

(2)

Wherein:

(3)

Formulas (2) and (3) for limiting probabilities were named Erlang formulas in honor of the founder of the queuing theory.

The QS failure probability is the limiting probability that all n channels of the system will be busy, i.e.

From here we find the relative throughput - the probability that the application will be served:

We obtain the absolute throughput by multiplying the intensity of the flow of requests λ by Q:

(4)

It remains only to find the average number of occupied channels k. This quantity could be found "directly", as the mathematical expectation of a discrete random variable with possible values \u200b\u200b0.1, ...,n and the probabilities of these valuesp 0, p 1, ..., p n:

Substituting here expressions (3) for pk and performing the appropriate transformations, we would finally get a formula for k. However, the average number of busy channels can be found easier if we take into account that the absolute throughputA the system is nothing but the flow rate of claims served by the system (per unit of time). Since each busy channel serves, on average, μ claims (per unit time), the average number of busy channels is:

or, given (4):


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In a Poisson stream of events (stationary and non-stationary), the number of events in the stream falling on any segment is distributed according to the Poisson law


Thus, for the system S under study with discrete states and continuous time, transitions from state to state occur under the action of Poisson streams of events with a certain intensity I.

We represent the car as some system S with discrete states iSj ,. 2. .... Sn, which passes from the state S / to the state Sj (i - 1, 2, ... .., n, j \u003d I, 2, ... .., and) under the influence of Poisson flows of events (failures) with intensities Xd. We will consider the following states of the car, in which it can be in operation and which are characterized by whole-day downtime

The Poisson stream of events is a stream that has two properties of ordinariness and no aftereffect.

In this section, a connection is established between Poisson streams of events and continuous time. It is shown how the intensity of Poisson stationary flows is used as the probability densities of system transitions from state to state in the analysis of models of specific situations.

There is a close connection between Poisson streams of events and discrete Markov processes with continuous time.

Connection of Poisson streams of events with discrete continuous-time Markov processes

That is, technically, it is easier to construct a Markov model with continuous time than a model with discrete time, although the problem of obeying the Poisson law of distribution of all flows of events that transfer system elements from state to state remains.

You can think of the events that move the car from state to state are streams of events (for example, streams of failures). If all flows of events that transfer the system (car) from state to state are Poisson (stationary or non-stationary), then the process taking place in the system will be Markov, and the probability densities of the transition Xy in a continuous Markov chain are the intensities of the flow of events that transfer the system from state Si to state Sj. For example, X03 is the rate of refusals of the car, which transfers the car from the healthy state, works to the state of being in TP.

Assumptions about the Poisson nature of the flow of events and about the exponential distribution of the time intervals between events are valuable in that they make it possible to apply the powerful apparatus of Markov random processes in practice.

Poisson stationary (simplest) stream of events

Poisson stationary (simplest) stream of events

Poisson unsteady flow of events

Consider an unsteady Poisson flow with intensity Mf), a certain time interval of length r\u003e 0, starting from the moment t0 (and ending, therefore, at the moment + r) and a discrete random variable X p r) - the number of events occurring in the flow during the time interval from ta to t0 + r.

Definition 6.2. The element of the probability of occurrence of an event in a non-stationary Poisson flow is the probability\u003e, (AO of the occurrence of an event in an elementary (sufficiently small) time interval from t0 to t0 + bt.

Theorem 6.2. For the element of the probability of occurrence of an event in an elementary time interval from t0 to t0 + Af in a nonstationary Poisson flow with intensity A (t), the approximate formula holds:

The main characteristic property of a nonstationary Poisson flow is that the probability of a certain number of events occurring in a time interval depends not only on its length, but also on the moment of its beginning.

One of the main stochastic characteristics of a nonstationary Poisson flow is a discrete random variable X (t т), which is a random number of events occurring in the flow during the interval [t. + T.

Another main stochastic characteristic of a nonstationary Poisson flow is a random time interval T (tB) between two neighboring events, the first of which occurred at time t0.

Proof The probability p (t At) that the system S, which was at the time moment t in the state sp during the time interval from t to t + Ы, will pass from it to the state s (see 4) is equal to the element of the probability pfa t) of the occurrence of the event in Poisson flow P .. on an elementary section from t to + D (see Definition 5.11). But (see (4.3))

The system, in which a discrete Markov process with continuous time takes place, jumps from one state x to another xj not spontaneously, but under the influence of a certain event, which we can attribute to the events of some Poisson flow P .. and thus assume that the transition of the system from state x to state x occurs under the influence of the entire flow / L. Attracting the entire flow P .. gives us the opportunity to consider the intensity A () of this flow.

Let us consider in more detail the case of Poisson demand distribution. The cost function will have a form similar to (5.6.18), with the replacement of integration over x by summation. Let us find the density 1\u003e (t) of the distribution of the deficit time. The distribution of the onset time of the kth event of the Poisson flow obeys the Erlang law of the kth order. The deficit begins when the entire stock S and one more unit are used up, so that

The total flow of failures associated with the entry of cars of the studied group into TO-2 is obtained by superposition (superposition) of the TO-2 flows of these cars. Calculations show that the distribution of the mileage interval between events in this stream obeys an exponential law. In this case, the TO-2 flow of all the vehicles under study is Poisson.

The image of the flow of refusals associated with car write-off is conditional. Indeed, if the car fails at the moment when the first event of a given flow occurs, then it does not matter at all whether the flow of failures continues after this or the fate of the car stops. In the case when the element (car) cannot be restored, the failure flow is Poisson.

Each of the aggregates included in the block is a complex system consisting of a large number of elements. Failure of each of them can lead to the loss of the ability to perform the assigned task of the entire unit. The flow of failures of the unit in time is formed as a result of the superposition of a set of events - flows of failures of the elements that make up its composition. When solving a practical problem, failures in elements can be considered as independent (or weakly dependent) and ordinary events, therefore, for the total flow of failures of the entire unit, it is legitimate to use the limit flow theorem in the theory of random processes. This theorem determines the conditions under which the sum of independent (or weakly dependent)

This flow occupies a central place among all the variety of flows, just like random variables with normal distribution in the applied theory of probability. This situation is explained by the fact that in the theory of flows, as well as in the theory of random variables, there is limit theorem, according to which the sum of a large number of independent flows with any distribution law approaches the simplest flow with an increase in the number of terms of flows.

Stationary poissonThe (simplest) flow is a flow that has three properties: ordinariness,no aftereffectand stationarity.

Distribution of events over a short time interval

By definition, the flow rate is called the limit
, since the simplest flow is stationary, then for it
.

The stationarity of the flow and the absence of aftereffect exclude the dependence of the probability of occurrence of events on the interval
both from the location of this interval on the time axis, and from the events preceding it. therefore
.

For any time interval we have. When striving
all the terms on the right-hand side of this formula, with the exception of the first, can be neglected, since due to the ordinariness of the flow of events, these values \u200b\u200bare negligible compared to
:

.

Taking into account the above, we transform the initial expression for the flow intensity:

.

Hence we have the equality
, i.e. the probability of occurrence of one event on a small time interval is proportional to this interval with the coefficient .

It's obvious that
... Hence,
, whence we have
- the probability of not appearing a single event on a small time interval
.

Distribution of events in a Poisson stream

Let's find the expression
where
is the probability that on the interval
will happen events. This event will occur in one of two mutually exclusive cases:

By the theorem of addition of the probabilities of inconsistent events, we have the probability of a situation 1 or 2:

Where from. By striving
, we get
.

Let us define a similar relation for
... For an event on the interval
did not come even once, it is necessary and sufficient for it to come 0 times in the interval and 0 once - in
... The probability of this event is. Whence we similarly obtain
.

Thus, the Poisson flow of events is described by the system of linear differential equations

,

with obvious initial conditions.

From the first equation we obtain
, from the initial conditions we have
from where c \u003d 1... Finally
.

Thus, for a Poisson flow, the probability
absenceevents on any interval of length is determined by exponential dependence. To solve the complete system of equations, we use the Laplace transform. We have,

from where
;
and further
;
; ...
.

Taking the inverse Laplace transform, using the tables, we obtain
, i.e. Poisson distribution.

Thus, the simplest flow obeys the Poisson distribution law, for which the mathematical expectation and variance are, respectively, equal
.

Distribution of intervals between events

Let's find the distribution law of time intervals between events for the simplest flow. Consider a random variable - the time interval between two arbitrary adjacent events in the simplest flow. It is required to find the distribution function
.

Consider the opposite event
... This is the probability that, starting at some point occurrence of the event, during no more events will appear. Since the flow has no aftereffect, the fact that the event appeared at the moment , should not have any effect on the behavior of the stream in the future. Therefore the probability
from where
and the probability density
.

This distribution law is called indicative(exponential) with parameter . Find the expected value and variance this process:

;

The exponential law has a remarkable property: if the period of time distributed according to the exponential law has already lasted for some time , then this does not in any way affect the distribution law of the remaining part of the interval
(it will be the same as the gap distribution law ).

Let us prove this property. Let
- the likelihood that the maintenance (c) will still last at least (c): i.e. on time interval a+ tnot a single event will happen. With the exponential distribution law of service time
.

By the theorem on the product of probabilities of events. With an exemplary law;
and therefore
, i.e. with the exponential law of service time, the distribution law for the remaining part of the service time does not depend on how long the service has already lasted. It can be proved that the exponential law onlyfor which this property is valid.

Considered propertyessentially represents a different formulation of the property no aftereffect.

It is customary to take the Poisson flow as the flow standard in modeling.

Poisson flow is an ordinary flow without aftereffect.

As previously indicated, the probability that over the time interval ( t 0 , t 0 + τ ) will happen m events, is determined from Poisson's law:

where a - Poisson parameter.

If λ (t) \u003d const ( t), that is stationary Poisson flow (simplest). In this case a = λ · t... If λ \u003d var ( t), that is unsteady Poisson flow.

For the simplest flow, the probability of occurrence m events during τ is equal to:

The probability of non-appearance (that is, none, m \u003d 0) events over time τ is equal to:

Figure: 28.2 illustrates dependence P 0 from time to time. Obviously, the longer the observation time, the less probability of non-occurrence of any event. Also, the more the value λ , the steeper the graph goes, that is, the faster the probability decreases. This corresponds to the fact that if the intensity of the occurrence of events is high, then the probability of non-occurrence of the event rapidly decreases with the time of observation.

The probability of occurrence of at least one event ( P HB1C) is calculated as follows:

as P HB1S + P 0 \u003d 1 (either at least one event will appear, or none will appear - no other is given).

From graph to fig. 28.3 it can be seen that the probability of the occurrence of at least one event tends to unity with time, that is, with the corresponding long-term observation of an event, such an event will necessarily occur sooner or later. The longer we watch the event (the more t), the more likely it is that the event will occur - the graph of the function monotonically increases.

The more the intensity of the occurrence of the event (the more λ ), the faster this event occurs, and the faster the function tends to unity. On the graph, the parameter λ represented by the slope of the line (slope of the tangent).

If you increase λ , then when observing the event during the same time τ , the probability of the occurrence of an event increases (see. fig. 28.4). Obviously, the graph starts from 0, because if the observation time is infinitely small, then the probability that the event will occur during this time is negligible. And vice versa, if the observation time is infinitely long, then the event will necessarily occur at least once, which means that the graph tends to a probability value equal to 1.

By studying the law, you can determine that: m x = 1/λ , σ = 1/λ , that is, for the simplest flow m x = σ ... The equality of the mathematical expectation to the standard deviation means that this flow is a flow without aftereffect. The dispersion (more precisely, the root mean square deviation) of such a flow is large. Physically, this means that the time of occurrence of an event (distance between events) is poorly predictable, by chance, is in the interval m x – σ < τ j < m x + σ ... Although it is clear that on average it is approximately equal to: τ j = m x = T n / N... The event can appear at any time, but within the range of this moment τ j relatively m x on [- σ ; +σ ] (aftereffect value). On fig. 28.5 the possible positions of event 2 relative to the time axis are shown for a given σ ... In this case, they say that the first event does not affect the second, the second on the third, and so on, that is, there is no aftereffect.

Within the meaning of P equally r (see lecture 23. Modeling a random event. Modeling a complete group of incompatible events), therefore, expressing τ from the formula (*) , finally to determine the intervals between two random events we have:

τ \u003d –1 / λ Ln ( r) ,

where r - evenly distributed from 0 to 1 random number, which is taken from the RNG, τ - the interval between random events (random variable τ j).

Example 1... Consider the flow of products arriving at a technological operation. Products arrive randomly - on average, eight pieces per day (flow rate λ \u003d 8/24 [units / hour]). It is necessary to simulate this process during T n \u003d 100 hours. m = 1/λ \u003d 24/8 \u003d 3, that is, on average, one part in three hours. notice, that σ \u003d 3. On fig. 28.6 an algorithm is presented that generates a stream of random events.

On fig. 28.7 the result of the algorithm is shown - the moments of time when the parts came to the operation. As you can see, in total for the period T n \u003d 100 production unit processed N \u003d 33 products. If we run the algorithm again, then N may turn out to be equal, for example, 34, 35 or 32. But on average, for K algorithm runs N will be equal to 33.33 ... If we calculate the distance between events t from i and points in time defined as 3 i, then on average the value will be equal to σ = 3.

The restored objects after repair continue to operate according to their intended purpose. The reliability of recoverable objects is usually assessed by the characteristics of the failure flow. IN general case streamevents is a sequence of homogeneous events following one after the other at random times. In the theory of reliability of restored objects, the simplest flows of events are mainly considered, characterized by ordinariness, stationarityand no aftereffect(these streams of events are most common in practice).

The stream of events is called ordinary,if the probability of occurrence of two or more failures in a unit time interval is negligible compared to the probability of occurrence of one failure. Thus, failures in the system occur one at a time.

The stream of events is called stationary,if the probability of a particular number of events falling within the time interval m depends only on the length of the interval and does not depend on where exactly this interval is located on the axis. The stationarity of the flow of events means that the flow density is constant. Obviously, when observed, the flow can have thickening and rarefaction. However, for a stationary flow, these thickening and rarefaction are not of a regular nature, and the average number of events falling within a unit time interval remains constant for the entire considered period.

No aftereffectin the simplest flow of events means that the probability of failures in a unit time interval does not depend on the occurrence of failures in all previous time intervals, i.e., failures occur independently of each other. In electronic computing facilities, the failure rate is equal to the sum of the failure rates of individual devices. If each individual flow has a fairly uniform and small effect on the total flow, then the total flow will be the simplest.

Let the simplest flow of failures have the following properties.

1. The time between failures is distributed exponentially with some parameter A, (formulas (4.16) - (4.21)):

Consequently, and T 0 - time to first failure is exponentially distributed with the same parameter X (mean time to first failure is the mathematical expectation T:

Under these conditions, the failure rate X (t) turns out to be a constant:

2. Let r (t) - number of refusals over time t (r (t) is a random variable). The likelihood that over time t will happen m failures at failure rate X, is determined by Poisson's law (see (4.22)):

3. Average number of bounces over time t equally:

4. The probability that over time t no failure will occur, is equal to: P (t) \u003d e~ and.

The described simplest flow of events is also called stationary Poisson flow.As mentioned above, such a flow is typical for complex highly reliable objects.

The process of functioning of the restored object can be described as a sequence of alternating intervals of operability and downtime associated with restoration. It is assumed that the object's failure is immediately recorded, and from the same moment the recovery procedure begins. Health intervals (we assume 100% recovery of the object) are independent and equally distributed random variables, and they do not depend on recovery intervals, which are also independent and equally distributed random variables (most likely with a different distribution). Each of these sequences of intervals forms its simplest stream of events.

Recall that in the case of recoverable objects, the main characteristic is failure flow parameter.The operation of such objects can be described as follows: at the initial moment of time, the object starts to work and works to failure, after a failure, recovery occurs and the object again works to failure, etc. leading functionQ (t) of a given flow, which is the mathematical expectation of the number of failures over time 1:

where r (t) - number of refusals over time t.

The failure flow parameter ω (0 characterizes the average number of failures expected in a short time interval, and is determined by the formula (2.9):

The leading function can be expressed in terms of the failure flow parameter:

For stationary Poisson flows,as mentioned above, the failure rate is a constant value and is equal to X; in this case, it coincides with the parameter of the flow of failures. Indeed, by property 3 of a stationary Poisson flow, the average number of failures in time r is equal to: Q. (t) = M \u003d Xt, hence,

Mean time between failures.As already mentioned, this indicator is the ratio of the operating time to the mathematical expectation of the number of failures during this operating time. Since with a steady flow of failures M)

 

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