Poisson process. Stationary Poisson failure flow Machine failures form a Poisson flow with intensity

Computer science, cybernetics and programming

Definition of Poisson flow. Poisson flow is an ordinary flow with no aftereffect. The classic traffic model in information networks is the Poisson simplest flow. It is characterized by a set of probabilities Pk of receipt of k messages for a time interval t: where k=01 is the number of messages; λ is the intensity of the flow.

1. Definition of Poisson flow. Properties.

Poisson flow is an ordinary flow with no aftereffect.

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

where k=0,1,… - number of messages; λ - flow intensity.

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 value of λ corresponds to a wider and more symmetrical probability density plot.

Rice. 1. Poisson distributions. Probability densities.

The mathematical expectation (mean) and the variance of the Poisson stream are λ t .

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

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

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

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

Rice. 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 mathematician A.K. Erlang. The problem is posed as follows: there are n channels (communication lines) that receive a flow of requests with intensity λ. The service flow of each channel has intensity μ. Find the limiting probabilities of the system states and indicators of its efficiency.

The system S (QS) has the following states (we number them according to the number of customers in the system): S 0 , S 1 ,…, S n , where S k the state of the system when there are k requests in it, i.e. k channels are occupied.

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

Rice. 3. QS state graph

The flow of requests sequentially transfers the system from any left state to the neighboring right one with the same intensity λ. The intensity of the flow of services that transfer the system from any right state to a neighboring left state is constantly changing depending on the state. Indeed, if the QS is in state S 2 (two channels are busy), then it can go to state S 1 (one channel is busy), when either the first or the second channel finishes servicing, i.e. the total intensity of their service flows will be 2μ . Similarly, the total flow of services that transfers the QS from the state S 3 (three channels busy) in S 2 , will have an intensity of 3μ , i.e. any of the three channels can become free, and so on.

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

(1)

where are the expansion terms- coefficients at p 0 in expressions for the limiting probabilities p 1 , p 2 ,..., p n .

Note that the intensities λ and μ do not enter into formula (1) separately, but only as the ratio μ/λ. Denote: μ/λ = p , and we will call the value ρ the reduced intensity of the flow of requests or the intensity of the channel load. It expresses the average number of requests arriving during the average service time of one request. Using this notation, we rewrite formula (1) in the form:

(2)

Wherein:

(3)

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

The QS failure probability is the marginal 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 applications λ by Q:

(4)

It remains only to find the average number of busy channels k. This value could be found "directly", as the mathematical expectation of a discrete random variable with possible values ​​0,1,..., n and the probabilities of these values p 0 , p 1 , …, p n :

Substituting here expressions (3) for p k and performing the appropriate transformations, we would eventually obtain a formula for k. However, the average number of occupied channels can be found more easily if we consider that the absolute throughput A system is nothing but the intensity of the flow of requests served by the system (per unit of time). Since each busy channel serves on average μ requests (per unit time), the average number of busy channels is:

or, given (4):


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In the Poisson flow of events (stationary and non-stationary), the number of events in the flow , falling on any site, 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 flows of events with a certain intensity H.

Let us represent a 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 = I, 2, ..., and) under the influence of Poisson event flows (failures) with intensities xd. We will consider the following states of the car, in which it can be in the process of operation and which are characterized by all-day downtime

The Poisson flow of events is a flow that has two properties - ordinary and no aftereffect.

In this section, a connection is established between Poisson event flows and with 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 relationship between Poisson event flows and continuous-time discrete Markov processes.

Relationship between Poisson event streams and continuous-time discrete Markov processes

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

We can consider that the events that move the car from state to state are streams of events (for example, failure streams). If all flows of events that transfer the system (car) from state to state are Poisson (stationary or non-stationary), then the process occurring in the system will be Markovian, and the probability densities of the transition Xu in a continuous Markov chain are the intensities of the flow of events that transfers the system from state Si to state Sj. For example, X03 is the intensity of the flow of car failures, which transfers the car from the state of serviceable, working to the state is in TR.

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

Poisson stationary (simplest) flow of events

Poisson stationary (simplest) flow of events

Poisson non-stationary event flow

Let us consider a non-stationary Poisson flow with intensity Mf), some time interval of length r>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 probability element of the occurrence of an event in a non-stationary Poisson flow is the probability >,(AO of the occurrence of an event for an elementary (sufficiently small) time interval from t0 to t0+bt.

Theorem 6.2. For the probability element of the occurrence of an event over an elementary time interval from t0 to t0 + Af in a non-stationary Poisson flow with intensity A(t), the approximate formula takes place

The main characteristic property of a non-stationary 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 non-stationary Poisson flow is a discrete random variable X(t t), which is a random number of events occurring in the flow over the interval [t.+t.

Another main stochastic characteristic of a non-stationary 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) of the fact that the system S, which was in the state sp at the time t in the time interval from t to t+Ы, will pass from it to the state s (see 4) is equal to the probability element pfa t) of the occurrence of an event in Poisson flow P.. on an elementary section from t to + A (see Definition 5.11). But (see (4.3))

A system in which a discrete Markov process with continuous time proceeds 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. Involvement of 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 the Poisson distribution of demand. The cost function will have a form similar to (5.6.18), with the integration over x replaced by summation. Let us find the density 1>(m) of the distribution of the shortage time. The distribution of the onset time of the kth event of the Poisson flow is subject to the Erlang law of the kth order. The shortage begins when the entire supply of S and one more unit is 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 superimposing (superposition) flows of TO-2 of these cars. Calculations show that the distribution of the run interval between events in this flow obeys the exponential law . At the same time, the TO-2 flow of all the vehicles under study is Poisson.

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

Each of the units 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 task of the entire unit. The flow of failures of the unit in time is formed as a result of the superposition of many events - the flows of failures of the elements included in 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 failure flow of the entire unit, it is legitimate to use the limit flow theorem in the theory of random processes. This theorem defines conditions under which the sum of independent (or weakly dependent)

This flow occupies a central place among the entire variety of flows, as well as random variables with a normal distribution law in applied probability theory. 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 the flows.

Stationary Poisson The (simplest) flow is a flow that has three properties: ordinary,lack of aftereffect and stationarity.

Distribution of events over a small time interval

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

The stationarity of the flow and the absence of an 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. That's why
.

For any time interval, we have . When striving
all members of the right side of this formula, with the exception of the first, can be neglected, because due to the ordinary nature of the flow of events, these quantities are negligible compared to
:

.

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

.

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

It's obvious that
. Consequently,
, whence we have
- the probability of not a single event occurring in a short time interval
.

Distribution of events in a Poisson stream

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

According to the theorem of addition of probabilities of incompatible events, we have the probability of occurrence of situation 1 or 2:

Where . striving
, we get
.

Let us define a similar relation for
. To event on the interval
never happened, it is necessary and sufficient for it to happen 0 times in the interval and 0 times - in
. The probability of this event is equal. Whence similarly we get
.

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

,

with obvious initial conditions .

From the first equation we get
, from the initial conditions we have
, where c = 1. Finally
.

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

where
;
and beyond
;
; ...
.

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

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

Distribution of intervals between events

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

Consider the opposite event
. This is the probability that, starting from some moment occurrence of the event, during no more events will appear. Since the flow is without 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
, where
and probability distribution density
.

This distribution law is called revealing(exponential) with parameter . Let's find the mathematical expectation and dispersion this process:

;

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

Let's prove this property. Let
- the probability that the maintenance continued (c), will still last at least (c): i.e. on the time interval a+ t no event will occur. Under the exponential law of service time distribution
.

According to the theorem on the product of the probabilities of events. Under the exponential law;
and hence
, i.e. under the exponential law of service time, the law of distribution of the remaining part of the service time does not depend on how much time the service has already lasted. It can be shown that the exponential law the only one for which this property is true.

Considered property, in essence, represents another formulation of the property lack of aftereffect.

For the flow standard in modeling, it is customary to take the Poisson flow.

Poisson flow is an ordinary flow with no aftereffect.

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

where a is the Poisson parameter.

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

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

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

Rice. 28.2 illustrates dependence P 0 from time. It is obvious that the longer the observation time, the lower the probability of no event occurring. Moreover, the higher 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 occurrence of events is high, then the probability of non-occurrence of an event decreases rapidly with the time of observation.

The probability that at least one event will occur ( P XB1C) is calculated as follows:

because P HB1S + P 0 = 1 (either at least one event will appear, or none will appear - the other is not given).

From the chart on rice. 28.3 it can be seen that the probability of the occurrence of at least one event tends to unity over time, that is, with an appropriate long-term observation of an event, one will certainly happen sooner or later. The longer we observe the event (the more t), the greater the probability that the event will occur - the graph of the function increases monotonically.

The greater 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 steepness of the line (the slope of the tangent).

If you increase λ , then when observing the event for the same time τ , the probability of the occurrence of the event increases (see rice. 28.4). Obviously, the graph starts from 0, since 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 definitely occur at least once, which means that the graph tends to a probability value of 1.

By studying the law, it can be determined 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 the given flow is a flow without aftereffect. The dispersion (more precisely, the standard deviation) of such a flow is large. Physically, this means that the time of occurrence of an event (the distance between events) is poorly predictable, random, 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. An event can appear at any moment of time, but within the range of this moment τ j relatively m x on the [- σ ; +σ ] (value of aftereffect). On the rice. 28.5 possible positions of event 2 relative to the time axis are shown for a given σ . In this case, we say that the first event does not affect the second, the second does not affect the third, and so on, that is, there is no aftereffect.

Within the meaning of P equals 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 = –1/ λ Ln( r) ,

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

Example 1. Consider the flow of products coming to the technological operation. Products arrive randomly - an average of eight pieces per day (flow rate λ = 8/24 [unit/hour]). It is necessary to simulate this process during T h = 100 hours. m = 1/λ = 24/8 = 3, that is, on average, one detail per three hours. notice, that σ = 3. On rice. 28.6 an algorithm generating a stream of random events is presented.

On the rice. 28.7 shows the result of the algorithm - the time points when the details came to the operation. As can be seen, in just the period T n = 100 production node processed N= 33 products. If we run the algorithm again, then N may be equal to, for example, 34, 35 or 32. But on average, for K algorithm runs N will be equal to 33.33 ... If we calculate the distances between events t With i and moments of time defined as 3 i, then the average value will be equal to σ = 3.

Restored objects after repair continue to be used for their intended purpose. Reliability of restored objects is usually evaluated by the characteristics of the flow of failures. AT general case flow events is a sequence of homogeneous events following one after another at random times. In the theory of reliability of restored objects, the simplest flows of events are mainly considered, characterized by ordinariness, stationarity and lack of aftereffect(Such event streams are most often encountered 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 that one or another number of events will fall into the time interval t depends only on the length of the interval and does not depend on where exactly on the axis this interval is located. The stationarity of the flow of events means that the density of the flow is constant. It is obvious that, when observed, the flow can have concentrations and rarefactions. However, for a stationary flow, these concentrations and rarefaction are not of a regular nature, and the average number of events falling on a unit time interval remains constant for the entire period under consideration.

No aftereffect in the simplest flow of events means that the probability of failures in a single 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 flow is equal to the sum of the failure flows of individual devices. If each individual flow has a sufficiently uniform and small effect on the total flow, then the total flow will be the simplest.

Let the simplest failure flow have the following properties.

1. The time between failures is distributed according to an exponential law with some parameter A, (formulas (4.16) - (4.21)):

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

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

2. Let r(t) - number of failures per time t (r(t) is a random variable). The probability that in time t happen m failure rate x, is determined by the Poisson law (see (4.22)):

3. Average number of failures per time t equals:

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

The simplest flow of events described 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 health and downtime associated with restoration. It is assumed that the failure of the object is immediately fixed and from the same moment the recovery procedure begins. Health intervals (we assume 100% recovery of the object) are independent and identically distributed random variables, while they do not depend on recovery intervals, which are also independent and identically distributed random variables (most likely with a different distribution). Each of these sequences of intervals forms its own simple stream of events.

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

where r(t) - number of failures per time t.

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

The leading function can be expressed in terms of the failure stream 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 failure flow parameter. Indeed, by property 3 of a stationary Poisson flow, the average number of failures in time r is: Q.(t) = M = Xt, Consequently,

MTBF. As already mentioned, this indicator is the ratio of 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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