12.37 Software Input/Output Information Guide Software Input/Output Information Guide explains to the user how to present, enter input information and how to interpret the output information, in what mode (batch or interactive) the system operates

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

To describe another important component of any system - the input flow of applications - a probabilistic law is usually specified, 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 a fairly long period of time. Sometimes there are systems in which applications can arrive 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 requests described by us (in our case, cars) is called Poisson).

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

The importance of scientific knowledge in technological development needs to be established. To perceive technology as the “application of scientific knowledge” means to perceive the latter as a phenomenon that occurs outside the functioning of technology as such. Here the focus is on the knowledge “inputs” (from science) that are important for production processes. This idea of ​​“received knowledge” conflicts with ample evidence that “technological improvements usually occur prior to their scientific understanding.”

Let's consider the conditions for the uninterrupted functioning of suppliers. They are expressed as constraints on the random input stream Qkl

Model a is intended 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 arriving at 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, healthy - faulty). The node model includes functions (tasks) to control the transformation of the flow passing through the node - functions of regulators, protections, interlocks.

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

Input Streams Process Output Streams

The presence of an incoming flow means the need to unload transport, check the quantity and quality of arriving cargo. The output flow determines the need to load vehicles, the internal flow determines the need to move cargo inside the warehouse.

Mixing flows. 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 of k-ro substance entering the system per unit time (molar flow). The mixing process is irreversible; entropy production can be found as the difference between the entropy of the output and input streams. Taking into account the constancy of their enthalpy, we obtain

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 in data files. These errors are the result of input data not meeting software specifications. In the internal section, such errors may appear in the form of constants or literals included in the code that initializes some calculations.

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

In addition to presenting mathematical schemes for simulation modeling, this chapter compares 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 when analytically modeling the QS of real objects, the simulation results never correspond to the behavior of the object, since they give the values ​​of the parameters of the QS in a steady state. Real objects that are modeled as a QS in a steady state, as a rule, are not found, 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 flows by 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 applications received on average per unit of time, and the law of probability distribution of the moments of arrival of applications into the system.

Synchronizing moments Fig. 5.1.1. Input flow of applications

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

Thus, for the simplest flows and elementary QSs, their qualitative parameters can be analytically calculated. Real economic objects, as a rule, represent complex QS systems 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 allows one 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 repeatedly reproducing random implementations of the process of arrival of requests and the process of servicing them under fixed conditions of the problem. By changing the conditions of the problem, parameters of input flows and elements of the QS, you can obtain the qualitative parameters of a given QS with certain changes. Qualitative parameters of the QS of the type listed above for the simplest input flows and elementary QS are assessed by statistical processing of values ​​that are qualitative indicators of the functioning of the QS.

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

In this example, we considered the simplest case of a Poisson input flow, exponential service time, one serving unit. 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 system that has become widespread to date. Kendall designated the type of queuing system using three symbols, the first of which describes the type of input flow, the second - the type of probabilistic description of the queuing system, and the third - the number of serving devices. The symbol M denoted the Poisson distribution of the input flow (with an exponential distribution of intervals between requests); the same symbol was used for the exponential distribution of service duration. Thus, the queuing system described and studied in this section is designated M/M/1. The M/G/3 system, for example, stands for a system with a Poisson input flow, a general (in English - general) service time distribution function and three service devices. There are also other notations D - deterministic distribution of intervals between arrivals of requests or service durations, E - Erlang distribution of order n, etc.  cost efficiency). And this requires a comprehensive examination, which is impossible without a scrupulous, in-depth and detailed analysis of the internal structure of the project, which allows you 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 interconnected elements, each of which can itself be considered a system.

However, there is one key component that was not taken into account in this analysis: productivity gains. Let us remember 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 input parameters. The overall productivity 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 inputs, resource reallocations and so on. When productivity rises, economic growth (GNP) can be greater than the increase in the difference between quantities flowing in (government spending and exports) and quantities flowing out (taxes and imports), because more output per unit of input creates new wealth at the aggregate level. As a consequence, it appears that Godley's arguments cannot be applied directly.

MGC-material flow supplying the logistics system (Input flow)

From the above relationships, we can draw the following conclusion for a given design of a binary distillation column, which determines the coefficients of heat and mass transfer, given compositions of flows at the inlet and outlet and the productivity of the column, steam consumption, reflux ratio and heat consumption supplied to the cube are fixed and can be found by the above ratios. If the compositions of only the input stream, one of the output streams, and the productivity of the target stream are specified, then the selection fraction (the concentration of the second output stream) can be selected, 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 a request received in a queuing system. This concept, depending on the specifics of the system, can have very different interpretations, for example, the number of devices , a communication line that receives incoming requests, a stacker crane that completes orders in a warehouse, etc. The random nature of the input flow of requests causes uneven loading K at some point in time they can be transferred

Definition 6.1. An input stream is called simplest if:

1) the probability of the appearance of a particular number of applications in a time interval depends only on its duration and does not depend on its location on the time axis (stationary input flow), and applications arrive one by one (singularity of the input flow) and independently of each other (no aftereffect during input stream);

2) the probability of the implementation of a separate random event (appearance of an application) on a time interval of short duration is proportional to an accuracy up to infinitesimal of a higher order of smallness compared to i.e. equal to where

3) the probability of the occurrence of two or more random events (the appearance of two or more applications) in a short time interval is the quantity

The absence of aftereffect in the definition of the simplest input flow means that for any non-overlapping time intervals, the number of applications arriving at one of these intervals does not depend on the number of applications arriving at other intervals.

Despite the fact that the input and output flows of many real queuing systems do not fully satisfy the definition of the simplest flow, the concept of the simplest flow is widely used in queuing theory. This circumstance is due not only to the fact that the simplest flows are quite common in practice, but also to the fact that the sum of an unlimited number of stationary ordinary flows with almost any aftereffect is the simplest flow. In this regard, let us consider the basic properties of the simplest flow.

Theorem 6.1. A discrete random variable that takes values ​​and characterizes, given the simplest input flow, the number of requests entering the service system on a time interval of duration t, is distributed according to Poisson’s law with the parameter

Let us consider a scalar random process with discrete states (i.e., for any fixed moment in time, its section ) is a discrete random variable with a set of possible values. Let its presence in the state mean the presence of k applications in the servicing system.

In accordance with the conditions of the theorem and the definition of the simplest flow, the random process , is a Markov homogeneous process with discrete states, and for any non-negative integer i and j, the probability density of the transition of the service system from state , to state at any time is determined by the equality

Therefore, in this case, the Kolmogorov system of equations has the following form:

where is the probability that on a time interval of duration t the service system under study will receive k requests. And since from Definition 6.1 of the simplest flow of requests it follows that

then we come to Cauchy problems for the function

and functions

Sequentially solving Cauchy problems (6.3), (6.4), in the case of the simplest input flow, we find the probability that the number of applications in a time interval of duration t will be equal to

Relations (6.5) mean that the random variable is distributed according to the Poisson law with the parameter

Corollary 6.1. If the input flow is the simplest, then the average number of requests entering the service system on a time interval of duration t is equal to

To determine the average number of applications, you need to find the mathematical expectation of a random variable. And since, according to (6.5), it is distributed according to Poisson’s law with the parameter then

According to the proven corollary, the parameter A represents the average number of applications received per unit of time. Therefore, it is called the intensity, or density of the simplest flow.

Corollary 6.2. If the input flow of requests is the simplest, then the dispersion of the scalar random variable characterizing the dispersion of the number of requests entering the queuing system on a time interval of duration t, relative to their average value, is equal to

M If the input flow is the simplest, then, according to (6.5), the random variable is distributed according to Poisson’s law with the parameter Therefore,

Let us pay attention to the fact that, according to (6.6) and (6.7), a random variable distributed according to Poisson’s law has the same mathematical expectation and variance.

Example 6.1. The service bureau receives an average of 12 orders per hour. Considering the flow of orders to be the simplest, we determine the probability that: a) not a single order will arrive in 1 minute; b) no more than three orders will arrive in 10 minutes.

Since the flow of orders is the simplest and the intensity is, according to (6.5), we have:

In accordance with Definition 6.1 of the simplest flow, the duration of the time interval between two successively arriving requests is a random variable. To build mathematical models of service systems, it is necessary to know the distribution function of a random variable or its distribution density (probabilities)

Theorem 6.2. In the case of the simplest input flow with intensity A, the duration of the time interval between two consecutive requests has an exponential distribution with parameter A.

Basic elements of a QS

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

The structure of the QS can be represented as follows:

Queuing system is a system that receives requests at random times requiring one type of service or another. In this case, when modeling a shopping center, the role of orders is played by buyers, and the role of devices is played by sellers.

Any system includes 4 main elements:

1) input stream

2) queue and service disciplines

3) device and service channel

4) output stream

Input stream

During operation, requests are received at the input of the servicing device at unknown times in advance, which are serviced for a certain random period of time, after which the device is released and can accept the next request. If a request arrives when the device is busy, it is denied service and gets queued. Due to the random nature of the flow of applications, at some points in time large queues may appear in the system, and at other times the system may operate underload or be completely idle. Therefore, the problems arise of quantitatively assessing the effectiveness of such systems, ensuring 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 streams are supplied to the input of the system, 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 order 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 a simple 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. Non-aftereffect means that events do not depend on the background. Singleness - all applications are processed one by one.

Queue and disciplines for servicing it

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

Queue-free QS are systems in which an incoming request is rejected if the service device is busy.

Queries with expectation can be limited or unrestricted by expectation. In systems with unlimited waiting, an incoming request will sooner or later be serviced. In systems with limited waiting, a number of restrictions are imposed on the time that applications stay in the system, regarding the time that applications stay in the queue, the time that applications stay in the system, etc.

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

1) discipline of filling the queue

2) the discipline of selecting applications from the queue

Queue filling disciplines include:

1) natural filling form

2) ring filling form

3) search form

4) priority filling form, with a shift in 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