By nature, the input flow of requirements is divided into a deterministic flow of requirements and a stochastic one (Fig. 2).

The deterministic input stream can be of two types. In the first case, the requests arrive at regular intervals. Another type of deterministic flow is a flow in which requirements arrive according to a known program - a schedule, when the moments of arrival of new requirements are known in advance.

Fig.2. Input stream classification

If the time intervals between receipts of claims are random, then this will be a stochastic process.

The stochastic demand flow is divided into three types: a flow with arbitrary stochastic properties, a recurrent flow, and a completely random or Poisson demand flow.

An arbitrary flow of requirements is characterized by the fact that it is not subject to any restrictions on the stochastic independence of the intervals between receipts of requirements, as well as on the nature of the probabilistic laws that describe the intervals between requirements.

An input stream is called recurrent if it is characterized by the following properties:

• the duration of the intervals between receipts of claims are stochastically independent;
• the duration of the intervals is described by the same distribution density.

An input stream is said to be completely random or simple if it is characterized by:

• the duration of the intervals between receipts of claims are statistically independent;
• the duration of the intervals is described by the same distribution density;
• the probability of receipt of claims on a sufficiently small interval Δt depends only on the value of Δt (this property is called stationarity or uniformity of arrival);
• the probability of receipt of requirements on the interval Δt does not depend on the history of the process;
• The nature of the demand flow is such that only one demand can arrive at any given time.

Thus, the simplest flow of requirements or a completely random flow is a flow that is determined by the properties of stationarity, ordinariness, and the absence of consequences at the same time.

The assumption of a completely random input flow of requirements is equivalent to the fact that the distribution density of time intervals between successive arrivals of requirements is described by an exponential law:

(1.1)

where λ is the intensity of requests coming into the system.

If the intervals are distributed exponentially, then the process is Poisson. Such processes are called M-processes (Markov processes).

In addition to Poisson's law, the Erlang distribution law is often used.

(1.2)

CMO with failures

A single-channel QS contains one channel (n = 1), and its input receives a Poisson flow of requests P in intensity (average number of events per unit time) of which in P in = λ. Since the intensity of the incoming flow can change over time, instead of λ, write λ (t). Then the time of service by the channel of one request T about is distributed according to the exponential law and is written as: , where λ is the failure rate.

The state of the QS is characterized by the idle or busyness of its channel, i.e. two states: S 0 - the channel is free and idle, S 1 - the channel is busy. The transition of the system from state S 0 to state S 1 is carried out under the influence of the incoming flow of requests P in, and from state S 1 to state S 0 the system is transferred by the service flow P o: if at a given time the system is in a certain after a given time, the QS goes into another state. The probability densities of the transition from the state S 0 to S 1 and back are equal to λ and µ, respectively. The state graph of such a QS with two possible states is shown in Fig.3.

Fig.3. Graph of states of a single-channel QS with failures.

For a multi-channel QS with failures (n > 1), under the same conditions, we denote the state of the system by the number of busy channels (by the number of requests under service in the system, since each channel in the QS is either free or serves only one request).

Thus, such QS can be in one of the following (n+1) states: s 0 - all n channels are free; s 1 - only one of the channels is occupied, the rest (n-1) channels are free; s i - busy i - channels, (n-i) channels are free; s n - all n channels are busy. The state graph of such a QS is shown in Fig.4.

Fig.4. Graph of states of a multichannel QS with failures.

In this case, there is a

Using the general rule for compiling Kolmogorov's differential equations, it is possible to compose systems of differential equations for the state graphs shown in Fig. 2 and Fig. 3:

for example, for a single-channel QS (Fig. 2) we have:

for a multichannel QS (Fig. 3), respectively, we have:

Having solved the first system of equations, one can find the values ​​of p 0 (t) and p 1 (t) for a single-channel QS and build graphs for three cases:

CMO with expectation

The queuing system has one channel. The incoming flow of service requests is the simplest flow with intensity λ. The intensity of the service flow is equal to µ (i.e., on average, a continuously busy channel will issue µ of serviced requests). Service duration is a random variable subject to an exponential distribution law. Service flow is the simplest Poisson flow of events. A request that arrives at a time when the channel is busy is queued and awaits service.

Let us assume that no matter how many requests enter the input of the serving system, this system (queue + served clients) cannot accommodate more than N-requirements (requests), i.e. customers who do not fall into the waiting period are forced to be served elsewhere. Finally, the source that generates service requests has an unlimited (infinitely large) capacity. The QS state graph in this case has the form shown in Fig.6.

Fig.6. Graph of states of a single-channel QS with expectation

QS states have the following interpretation:

S 0 - the channel is free;

S 1 - the channel is busy (there is no queue);

S 2 - the channel is busy (one request is in the queue);

S n - the channel is busy (n-1 requests are in the queue);

S N - the channel is busy (N-1 requests are in the queue).

The stationary process in this system will be described by the following system of algebraic equations:

(1.11)

where ρ=λ/µ; n - state number.

The solution of the above system of equations (1.10) for our QS model has the form:

(1.12)

(1.13)

It should be noted that the fulfillment of the stationarity condition for this QS is not necessary, since the number of applications admitted to the serving system is controlled by introducing a restriction on the queue length (which cannot exceed N-1), and not by the ratio between the intensities of the input stream, i.e. not by the relation λ/µ=ρ. Let us define the characteristics of a single-channel QS with waiting and a limited queue length equal to (N-1): the probability of refusal to service the request:

(1.14)

relative system throughput:

(1.15)

absolute bandwidth:

average number of applications in the system:

(1.17)

Average residence time of an application in the system:

average duration of stay of the client (application) in the queue:

(1.19)

average number of applications (clients) in the queue (queue length):

. (1.20) .

Now let's consider in more detail a QS that has n-channels with an unlimited queue. The flow of requests arriving at the QS has an intensity λ, and the flow of services has an intensity µ. It is necessary to find the limiting probabilities of QS states and indicators of its effectiveness.

The system can be in one state S 0 , S 1 , S 2 ,…,S k ,…,S n ,…, numbered according to the number of requests in the QS: S0 - there are no requests in the system (all channels are free); S 1 - one channel is busy, the rest are free; S 2 - two channels are occupied, the rest are free; …, S k - k channels are occupied, the rest are free; …, S n - all n channels are occupied (there is no queue); S n +1 - all n channels are occupied, there is one request in the queue; …, S n + r - all n channels are occupied, r requests are in the queue, ….

average number of applications in the queue:

(1.32)

average number of applications in the system:

(1.31) .

With every stretch of time a,a+T], let us associate a random variable X, equal to the number of requests received by the system during the time T.

The flow of requests is called stationary, if the distribution law does not depend on the starting point of the interval A, but depends only on the length of the given interval T.

For example, the flow of applications to the telephone exchange during the day ( T\u003d 24 hours) cannot be considered stationary, but from 13 to 14 hours ( T\u003d 60 minutes) - you can.

The flow is called no aftereffect, if the history of the flow does not affect the receipt of requirements in the future, i.e. requirements are independent of each other.

The flow is called ordinary, if no more than one request can enter the system in a very short period of time.

For example, the flow to the hairdresser is ordinary, but not to the registry office. But if as a random variable X consider pairs of applications entering the registry office, then such a flow will be ordinary (i.e., sometimes an extraordinary flow can be reduced to an ordinary one).

The flow is called the simplest , if it is stationary, without aftereffect and ordinary.

Main theorem . If the flow is the simplest, then the r.v. X is distributed according to the Poisson law, i.e. .

Corollary 1. The simplest flow is also called Poisson flow.

Corollary 2. M(X)=M(X[a,a+T] )=lt, i.e. during T enters the system on average lT applications. Therefore, for one unit of time, the system receives on average l applications. This value is called intensity input stream.

When solving command and control tasks, including command and control of troops, a number of tasks of the same type often arise:

• assessment of the throughput of a communication direction, a railway junction, a hospital, etc.;
• assessment of the effectiveness of the repair base;
• determination of the number of frequencies for the radio network, etc.

All these tasks are of the same type in the sense that they have a massive demand for service. A certain set of elements is involved in meeting this demand, forming a queuing system (QS) (Fig. 2.9).

The elements of the CMO are:

• input (incoming) demand flow(applications) for service;
• service devices (channels);
• queue of applications awaiting service;
• day off ( outgoing) flow serviced requests;
• flow of unserved requests;
• queue of free channels (for multichannel QS).

Incoming stream is a collection of service requests. Often the application is identified with its carrier. For example, the flow of faulty radio equipment entering the union workshop is a flow of applications - requirements for service in this QS.

As a rule, in practice, they deal with the so-called recurrent streams, streams that have the following properties:

• stationarity;
• ordinary;
• limited aftereffect.

We defined the first two properties earlier. As for the limited aftereffect, it lies in the fact that the intervals between incoming applications are independent random variables.

There are many recurrent streams. Each interval distribution law generates its own recurrent flow. Recurrent streams are otherwise known as Palm streams.

A flow with a complete absence of aftereffect, as already noted, is called a stationary Poisson flow. Its random intervals between requests have an exponential distribution:

here is the intensity of the flow.

The name of the stream - Poisson - comes from the fact that for this flow probability the appearance of applications for the interval is determined by the Poisson law:

A flow of this type, as noted earlier, is also called the simplest. It is this flow that designers assume when developing a QS. This is due to three reasons.

Firstly, a flow of this type in queuing theory is similar to the normal distribution law in probability theory in the sense that the passage to the limit for a flow that is the sum of flows with arbitrary characteristics with an infinite increase in terms and a decrease in their intensity leads to the simplest flow. That is, the sum of arbitrary independent (without predominance) flows with intensities is the simplest flow with intensity

Secondly, if the serving channels (devices) are designed for the simplest flow of requests, then the servicing of other types of flows (with the same intensity) will be provided with no less efficiency.

Third, it is this flow that determines the Markov process in the system and, consequently, the simplicity of the analytical analysis of the system. For other flows, the analysis of QS functioning is complicated.

Often there are systems in which the flow of input requests depends on the number of requests that are in service. Such SMOs are called closed(otherwise - open). For example, the work of a communication workshop of an association can be represented by a closed-loop QS model. Let this workshop be designed to service the radio stations that are in the association. Each of them has failure rate. The input stream of the failed equipment will have intensity:

where is the number of radio stations already in the workshop for repairs.

Applications may have different rights to start service. In this case, the applications are said to be heterogeneous. The advantages of some application streams over others are given by the priority scale.

An important characteristic of the input stream is the coefficient of variation:

where is the mathematical expectation of the length of the interval;

Standard deviation of a random variable (interval length).

For the simplest flow

For most real threads.

When the flow is regular, deterministic.

The coefficient of variation- a characteristic that reflects the degree of uneven receipt of applications.

Service channels (devices). A QS may have one or more service devices (channels). According to this QS are called single-channel or multi-channel.

Multichannel QS can consist of the same type or different types of devices. Service devices can be:

• communication lines;
• masters of repair bodies;
• runways;
• vehicles;
• berths;
• barbers, vendors, etc.

The main characteristic of the channel is the service time. As a rule, service time is a random value.

Usually, practitioners assume that the service time has an exponential distribution law:

where - service intensity, ;

Mathematical expectation of service time.

That is, the service process is Markovian, and this, as we now know, provides significant convenience in analytical mathematical modeling.

In addition to the exponential, there are -Erlang distribution, hyperexponential, triangular and some others. This should not confuse us, since it is shown that the value of the QS efficiency criteria does not depend much on the form of the service time probability distribution law.

In the study of QS, the essence of service falls out of consideration, quality of service.

Channels can be absolutely reliable, that is, do not fail. Rather, it can be accepted in the study. Channels may have ultimate reliability. In this case, the QS model is much more complicated.

Application queue. Due to the random nature of the flows of requests and services, an incoming request may find the channel (s) busy servicing the previous request. In this case, it will either leave the QS unserved, or remain in the system, waiting for the start of its service. Accordingly, there are:

• CMO with failures;
• CMO with expectation.

CMO with expectation are characterized by the presence of queues. A queue can have a limited or unlimited capacity: .

Researchers are usually interested in the following statistical characteristics associated with the stay of applications in the queue:

• the average number of applications in the queue for the study interval;
• average time of stay (waiting) of the application in the queue. QS with limited queue capacity referred to as a mixed-type SMO.

Often there are CMOs in which applications have limited time in line regardless of its capacity. Such QSs are also referred to as mixed-type QSs.

Outgoing stream is the flow of serviced requests leaving the QS.

There are cases when applications pass through several QS: transit connection, production pipeline, etc. In this case, the outgoing stream is incoming for the next QS. A set of sequentially interconnected QS is called multiphase QS or CMO networks.

The incoming stream of the first QS, having passed through the subsequent QS, is distorted and this makes modeling difficult. However, it should be borne in mind that with the simplest input stream and exponential service (that is, in Markov systems), the output stream is also the simplest. If the service time has a non-exponential distribution, then the outgoing stream is not only not simple, but also not recurrent.

Note that the intervals between outgoing requests are not the same as service intervals. After all, it may turn out that after the end of the next service, the QS is idle for some time due to the lack of applications. In this case, the outgoing flow interval consists of the idle time of the QS and the service interval of the first claim that arrived after the downtime.

The main elements of the CMO

The shopping center is a single-phase multichannel system with one queue of finite length. When the queue is full, the application is rejected. The purpose of solving the modeling problem is to determine the optimal number of service devices so that the average time the customer stays in the system does not exceed the specified one.

The structure of the QS can be represented as follows:

queuing system A system is called a system to which, at random times, requests arrive that need one or another type of service. In this case, when modeling a shopping center, buyers play the role of requests, and sellers play the role of appliances.

Any system includes 4 main elements:

1) input stream

2) queue and service disciplines

3) device and service channel

4) output stream

input stream

In the process of operation, requests are received at the input of the servicing device at unknown moments of time, which are serviced for a certain random period of time, after which the device is released and can accept the next request. If the application arrives when the device is busy, then it receives a denial of service and gets in the queue. Due to the random nature of the flow of requests, large queues may arise in the system at some points in time, and at other times the system may work with underload or even be idle. Therefore, there are problems of quantitative evaluation of the effectiveness of such systems, which ensure the minimization of the total costs associated with waiting and losses on the part of service facilities.

The input stream can be one-dimensional or multidimensional. If several different flows are fed to the system input, then it is multidimensional. Any input stream is represented by a sequence of homogeneous events following one after another at random times. The interval between two events is called the request arrival interval.

If the interval of receipt of applications is a random variable, i.e. changes according to a random distribution law, then the flow is called random.

A flow is called the simplest or stationary Poisson flow if it has 3 properties:

1) stationarity

2) no aftereffect

3) ordinary

Stationarity means that all probabilistic characteristics of the flow do not depend on time. No aftereffect means that events do not depend on history. Ordinariness - all applications pass one by one.

Queue and disciplines of its service

A queue is understood as a linear chain, lined up in a series of applications in a particular type of service. Depending on the presence of a queue, QS are divided into systems without a queue and systems with waiting.

QS without a queue are systems in which an incoming request is rejected if the service device is busy.

QS with expectation can be limited and unlimited expectation. In systems with unlimited waiting, the incoming request will be serviced sooner or later. In systems with limited waiting, a number of restrictions are imposed on the residence time of applications in the system, regarding the residence time of applications in the queue, the residence time of applications in the system, etc.

To regulate and coordinate the work of the queue, the following disciplines are used:

1) queue filling discipline

2) the discipline of choosing applications from the queue

Queue filling disciplines include:

1) natural fill shape

2) ring shape filling

3) search form

4) priority filling form, with a shift of other applications

Disciplines for selecting applications from the queue include 3 types:

1) first come first served

2) last come first served

3) selection of applications by priority

L () - the input stream of objects to be detected - the intensity of search efforts

To describe the other most important component of any , - the input flow of applications, - they usually set a probabilistic law, which is satisfied by the duration of the intervals between two successively arriving applications. These durations are usually statistically independent and their distribution does not change over some sufficiently long period of time. Sometimes there are systems in which requests can come in groups (for example, visitors to a cafe). It is generally assumed that the source from which applications are received is practically

Poisson distribution, therefore the input flow of applications described by us (in our case, cars) is called Poisson).

Here aa, c - vectors A, G, C - coefficient matrices y x - vectors of output and input flows of the object and - vector of variables that provide the range dependence of outputs on inputs.

It is necessary to establish the importance of scientific knowledge in technological development. To perceive technology as "the application of scientific knowledge" means to perceive the latter as a phenomenon that occurs outside the framework of the functioning of technology as such. Here attention is concentrated on the "input streams" of knowledge (from science) that are important for production processes. This notion of "knowledge gained" is in conflict with the abundant evidence that "technological improvements usually occur without scientific understanding."

Consider the conditions for the uninterrupted functioning of suppliers. They are expressed as constraints on the random input stream Qkl

Model a is designed to represent the structure of the unit (node) in the TP and simulate its operation by changing the states of the life cycle as a function of commands and events received by it. In this case, the life cycle states represent the operations performed by the node on the input stream and the state of the node (busy - free, good - faulty). The node model includes the functions (tasks) of managing the transformation of the flow passing through the node - the functions of regulators, protections, and blocking.

The diagram depicts the three main input streams (water, food and fuel) and the three output streams (sewage, solid waste and air pollution) that are common to all cities. In this model, quantities appear measured in natural units, namely, production waste for each type of pollutant. This circumstance significantly changes the usual properties of the input-output model, in which all values ​​are expressed in cost form.

Input streams Process Output streams

The presence of an input stream means the need to unload vehicles, check the quantity and quality of the cargo that has arrived. The output stream necessitates the loading of transport, the internal one - the need to move cargo inside the warehouse.

Mixing streams. Let us first consider the case when flows of pure substances having the same temperature T are mixed in the system. Let us denote by Nk the number of moles k-ro of the substance entering the system per unit time (molar flow rate). The mixing process is irreversible, the entropy production can be found as the difference between the entropy of the output and input streams. Taking into account the invariance of their enthalpy, we obtain

The function (p depends, like F in expression (1.79), on the parameters of the input stream and the stream enriched with the target component

Since p

Error values ​​contain constants and literals. In the input section, similar errors occur in user input streams and data files. These errors are the result of input data not meeting program specifications. In the internal section, such errors may appear as constants or literals that are part of the code that initializes some calculations.

The work of an accountant user in solving problems consists in performing repetitive technological operations (commands) on a PC, implemented in the active dialogue mode by typing commands on the keyboard, or in automatic (program) mode, in which the input command stream is pre-formed into a special program (command file). In the active dialogue mode, various tasks that are not predictable in advance are solved, various reference, analytical and other information is issued upon request and as needed.

In addition to the presentation of mathematical simulation schemes, this chapter compares the analytical and simulation modeling of QS from the standpoint of adequacy to the modeled object. As a result of this comparison, an important conclusion arises that in the analytical modeling of QS of real objects, the simulation results never correspond to the behavior of the object, since they give the values ​​of the QS parameters in the steady state. Real objects that are modeled in the form of a QS, as a rule, are not in a steady state, since the input flows and the QS themselves are constantly changing their parameters and distributions, and therefore, the QS is always in a transitional mode. Only simulation modeling of the QS, which does not limit the input streams with the requirements of stationarity, homogeneity, lack of

The input flow of applications (requirements for service) is characterized by a certain organization and a number of parameters (Fig. 5.1.1) by the intensity of receipt of applications, i.e. the number of requests, on average, received per unit of time, and the law of distribution of probabilities of the moments of arrival of requests in the system.

Synchronizing moments Fig. 5.1.1. Input stream of applications

Let us consider in more detail the characteristics of the input flow of applications and the simplest QS. The flow of homogeneous events is the time sequence of the appearance of requests for service, provided that all requests are equal. There are also streams of heterogeneous events, when one or another application has some kind of priority.

Thus, for the simplest flows and elementary QS, one can analytically calculate their qualitative parameters. Real economic objects, as a rule, represent complex QS both in structure and in input flows and parameters. In most cases, analytical expressions for assessing the quality of QS that model real economic objects and processes cannot be found. The application of the simulation method to queuing problems makes it possible to find the necessary quality indicators for economic systems of any complexity, if it is possible to construct algorithms for simulating each part of the QS.

The operation of the algorithm consists in multiple reproduction of random realizations of the process of arrival of requests and the process of their servicing under fixed conditions of the problem. By changing the conditions of the problem, the parameters of the input flows and QS elements, it is possible to obtain the qualitative parameters of this QS with certain changes. Qualitative parameters of the QS of the type listed above for the simplest input streams and elementary QS are estimated by statistical processing of quantities that are qualitative indicators of QS functioning.

This distribution is commonly called the Poisson distribution, therefore the input stream of applications described by us (in our case, cars) is called Poisson. We are not going to present here the derivation of formulas (2.1) and (2.2), the reader will find it in the book of Gnedenko B.V., A Course in Probability Theory. - M. Science, 1969.

In this example, we have considered the simplest case of Poisson input, exponential service time, one service setup. In fact, in reality, distributions are much more complicated, and gas stations include a larger number of gas stations. In order to streamline the classification of queuing systems, the American mathematician D. Kendall proposed a convenient notation that has become widespread by now. The type of queuing system Kendall designated using three symbols, the first of which describes the type of input stream, the second - the type of probabilistic description of the queuing system, and the third - the number of serving devices. He used the symbol M to denote the Poisson distribution of the input stream (with an exponential distribution of intervals between requests), the same symbol was also used for the exponential distribution of service duration. Thus, the queuing system described and studied in this section has the designation M/M/1. The M/G/3 system, for example, stands for a system with a Poisson input stream, a general (in English - general) service time distribution function, and three serving devices. There are also other designations D - deterministic distribution of intervals between the arrival of requests or service times, E - Erlang distribution of order n, etc. (cost efficiency). And this requires a comprehensive examination, which is impossible without a rigorous, deep and detailed analysis of the internal structure of the project, which allows to calculate the costs incurred and calculate (describe) the expected benefits. Then the project ceases to be a "black box", but is considered as an economic system. An economic system is usually understood as a complex of interrelated elements, each of which can itself be considered as a system.

However, there is one key component that performance gains were not included in this analysis. Recall that labor productivity is defined as the real output produced per hour of work. Similarly, the total productivity factor is defined as the real output per unit of the totality of all inputs. The overall performance factor reflects, in part, the overall efficiency with which inputs are converted into outputs. This is often associated with technology, but also reflects the impact of many other factors such as economies of scale, any unaccounted for inputs, reallocations of resources, and so on. When productivity rises, the growth of the economy (GNP) can be greater than the increase in the difference between inflows (government spending and exports) and outflows (taxes and imports), because more output per unit of input creates new wealth at the aggregate level. As a consequence, it seems that Godley's arguments cannot be applied directly.

MHZ - material flow that supports the logistics system (Input flow)

From the above relations, the following conclusion can be drawn for a given design of a binary distillation column, which determines the coefficients of heat and mass transfer, given compositions of inlet and outlet flows, and column performance, steam consumption, reflux ratio, and heat costs supplied to the cube are fixed and can be found from the above ratios. If the compositions of only the input stream, one of the outlet streams and the productivity of the target stream are given, then the selection fraction (concentration of the second stream at the outlet) can be chosen, minimizing the energy costs for separation.

CHANNEL (service) (hannel, server) - one of the fundamental concepts of queuing theory, denoting a functional element that directly fulfills an application that has entered the queuing system. This concept, depending on the specifics of the system, can have a variety of interpretations, for example, a device , a communication line that accepts incoming requests, a stacker crane that completes orders in a warehouse, etc. The random nature of the input flow of applications causes uneven loading K at some point in time