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Construction of a one-parameter, stochastic model of the production process
Ph.D. Assoc. Mordasov Yu.P.
University of Mechanical Engineering, 8-916-853-13-32, [email protected] gi
Annotation. The author has developed a mathematical, stochastic model of the production process, depending on one parameter. The model has been tested. For this, a simulation model of the production, machine-building process was created, taking into account the influence of random disturbances-failures. Comparison of the results of mathematical and simulation modeling confirms the expediency of applying the mathematical model in practice.
Key words: technological process, mathematical, simulation model, operational control, approbation, random perturbations.
The costs of operational management can be significantly reduced by developing a methodology that allows you to find the optimum between the costs of operational planning and the losses that result from the discrepancy between planned indicators and indicators of real production processes. This means finding the optimal duration of the signal in the feedback loop. In practice, this means a reduction in the number of calculations of calendar schedules for launching assembly units into production and, due to this, saving material resources.
The course of the production process in mechanical engineering is probabilistic in nature. The constant influence of continuously changing factors does not make it possible to predict for a certain perspective (month, quarter) the course of the production process in space and time. In statistical scheduling models, the state of a part at each specific point in time should be given in the form of an appropriate probability (probability distribution) of its being at different workplaces. However, it is necessary to ensure the determinism of the final result of the enterprise. This, in turn, implies the possibility, using deterministic methods, to plan certain terms for parts to be in production. However, experience shows that various interconnections and mutual transitions of real production processes are diverse and numerous. When developing deterministic models, this creates significant difficulties.
An attempt to take into account all the factors that affect the course of production makes the model cumbersome, and it ceases to function as a tool for planning, accounting and regulation.
A simpler method for constructing mathematical models of complex real processes that depend on a large number of different factors, which are difficult or even impossible to take into account, is the construction of stochastic models. In this case, when analyzing the principles of functioning of a real system or when observing its individual characteristics, probability distribution functions are built for some parameters. In the presence of high statistical stability of the quantitative characteristics of the process and their small dispersion, the results obtained using the constructed model are in good agreement with the performance of the real system.
The main prerequisites for building statistical models of economic processes are:
Excessive complexity and associated economic inefficiency of the corresponding deterministic model;
Large deviations of the theoretical indicators obtained as a result of the experiment on the model from the indicators of actually functioning objects.
Therefore, it is desirable to have a simple mathematical apparatus that describes the impact of stochastic disturbances on the global characteristics of the production process (commercial output, volume of work in progress, etc.). That is, to build a mathematical model of the production process, which depends on a small number of parameters and reflects the total influence of many factors of a different nature on the course of the production process. The main task that a researcher should set himself when building a model is not passive observation of the parameters of a real system, but the construction of such a model that, with any deviation under the influence of disturbances, would bring the parameters of the displayed processes to a given mode. That is, under the action of any random factor, a process must be established in the system that converges to a planned solution. At present, in automated control systems, this function is mainly assigned to a person, who is one of the links in the feedback chain in the management of production processes.
Let us turn to the analysis of the real production process. Usually, the duration of the planning period (the frequency of issuing plans to workshops) is selected based on the traditionally established calendar time intervals: shift, day, five days, etc. They are guided mainly by practical considerations. The minimum duration of the planning period is determined by the operational capabilities of the planned bodies. If the production and dispatching department of the enterprise copes with the issuance of adjusted shift tasks to the shops, then the calculation is made for each shift (that is, the costs associated with the calculation and analysis of planned targets are incurred every shift).
To determine the numerical characteristics of the probability distribution of random
A series of "Economics and Management" disturbances will build a probabilistic model of a real technological process of manufacturing one assembly unit. Here and hereinafter, the technological process of manufacturing an assembly unit means a sequence of operations (works for the manufacture of these parts or assemblies), documented in the technology. Each technological operation of manufacturing products in accordance with the technological route can be performed only after the previous one. Consequently, the technological process of manufacturing an assembly unit is a sequence of events-operations. Under the influence of various stochastic reasons, the duration of an individual operation may change. In some cases, the operation may not be completed during the validity of this shift job. It is obvious that these events can be decomposed into elementary components: performance and non-performance of individual operations, which can also be put in correspondence with the probabilities of performance and non-performance.
For a specific technological process, the probability of performing a sequence consisting of K operations can be expressed by the following formula:
PC5 \u003d k) \u003d (1-pk + 1) PG \u003d 1P1, (1)
where: P1 - the probability of performing the 1st operation, taken separately; r is the number of the operation in order in the technological process.
This formula can be used to determine the stochastic characteristics of a specific planning period, when the range of products launched into production and the list of works that must be performed in a given planning period, as well as their stochastic characteristics, which are determined empirically, are known. In practice, only certain types of mass production, which have a high statistical stability of characteristics, satisfy the listed requirements.
The probability of performing one single operation depends not only on external factors, but also on the specific nature of the work performed and on the type of assembly unit.
To determine the parameters of the above formula, even with a relatively small set of assembly units, with small changes in the range of manufactured products, a significant amount of experimental data is required, which causes significant material and organizational costs and makes this method for determining the probability of uninterrupted production of products hardly applicable.
Let us subject the obtained model to the study for the possibility of its simplification. The initial value of the analysis is the probability of failure-free execution of one operation of the technological process of manufacturing products. In real production conditions, the probabilities of performing operations of each type are different. For a specific technological process, this probability depends on:
From the type of operation performed;
From a specific assembly unit;
From products manufactured in parallel;
from external factors.
Let us analyze the influence of fluctuations in the probability of performing one operation on the aggregated characteristics of the production process of manufacturing products (the volume of commercial output, the volume of work in progress, etc.) determined using this model. The aim of the study is to analyze the possibility of replacing in the model of various probabilities of performing one operation with an average value.
The combined effect of all these factors is taken into account when calculating the average geometric probability of performing one operation of the averaged technological process. An analysis of modern production shows that it fluctuates slightly: practically within 0.9 - 1.0.
A clear illustration of how low the probability of performing one operation
walkie-talkie corresponds to a value of 0.9, is the following abstract example. Let's say we have ten pieces to make. The technological processes of manufacturing each of them contain ten operations. The probability of performing each operation is 0.9. Let us find the probabilities of lagging behind the schedule for a different number of technological processes.
A random event, which consists in the fact that a specific technological process of manufacturing an assembly unit will fall behind the schedule, corresponds to the underperformance of at least one operation in this process. It is the opposite of an event: the execution of all operations without failure. Its probability is 1 - 0.910 = 0.65. Since schedule delays are independent events, the Bernoulli probability distribution can be used to determine the probability of schedule delay for a different number of processes. The calculation results are shown in Table 1.
Table 1
Calculation of the probabilities of lagging behind the schedule of technological processes
to C^o0.35k0.651O-k Sum
The table shows that with a probability of 0.92, five technological processes will fall behind the schedule, that is, half. The mathematical expectation of the number of technological processes lagging behind the schedule will be 6.5. This means that, on average, 6.5 assembly units out of 10 will lag behind the schedule. That is, on average, from 3 to 4 parts will be produced without failures. The author is unaware of examples of such a low level of labor organization in real production. The considered example clearly shows that the imposed restriction on the value of the probability of performing one operation without failures does not contradict practice. All of the above requirements are met by the production processes of machine-assembly shops of machine-building production.
Thus, to determine the stochastic characteristics of production processes, it is proposed to construct a probability distribution for the operational execution of one technological process, which expresses the probability of performing a sequence of technological operations for manufacturing an assembly unit through the geometric average probability of performing one operation. The probability of performing K operations in this case will be equal to the product of the probabilities of performing each operation, multiplied by the probability of not performing the rest of the technological process, which coincides with the probability of not performing the (K + T)-th operation. This fact is explained by the fact that if any operation is not performed, then the following ones cannot be executed. The last entry differs from the rest, as it expresses the probability of complete passage without failure of the entire technological process. The probability of performing K of the first operations of the technological process is uniquely related to the probability of not performing the remaining operations. Thus, the probability distribution has the following form:
PY=0)=p°(1-p),
Р(§=1) = р1(1-р), (2)
P(^=1) = p1(1-p),
P(t=u-1) = pn"1(1 - p), P(t=n) = pn,
where: ^ - random value, the number of performed operations;
p is the geometric mean probability of performing one operation, n is the number of operations in the technological process.
The validity of the application of the obtained one-parameter probability distribution is intuitively evident from the following reasoning. Let's assume that we have calculated the geometric mean of the probability of performing one 1 operation on a sample of n elements, where n is large enough.
p = USHT7P7= tl|n]t=1p!), (3)
where: Iy - the number of operations that have the same probability of execution; ] - index of a group of operations that have the same probability of execution; m - the number of groups consisting of operations that have the same probability of execution;
^ = - - relative frequency of occurrence of operations with the probability of execution p^.
According to the law of large numbers, with an unlimited number of operations, the relative frequency of occurrence in a sequence of operations with certain stochastic characteristics tends in probability to the probability of this event. Whence it follows that
for two sufficiently large samples = , then:
where: t1, t2 - the number of groups in the first and second samples, respectively;
1*, I2 - the number of elements in the group of the first and second samples, respectively.
It can be seen from this that if the parameter is calculated for a large number of tests, then it will be close to the parameter P calculated for this rather large sample.
Attention should be paid to the different proximity to the true value of the probabilities of performing a different number of process operations. In all elements of the distribution, except for the last one, there is a factor (I - P). Since the value of the parameter P is in the range of 0.9 - 1.0, the factor (I - P) fluctuates between 0 - 0.1. This multiplier corresponds to the multiplier (I - p;) in the original model. Experience shows that this correspondence for a particular probability can cause an error of up to 300%. However, in practice, one is usually interested not in the probabilities of performing any number of operations, but in the probability of complete execution without failures of the technological process. This probability does not contain a factor (I - P), and, therefore, its deviation from the actual value is small (practically no more than 3%). For economic tasks, this is a fairly high accuracy.
The probability distribution of a random variable constructed in this way is a stochastic dynamic model of the manufacturing process of an assembly unit. Time participates in it implicitly, as the duration of one operation. The model allows you to determine the probability that after a certain period of time (the corresponding number of operations) the production process of manufacturing an assembly unit will not be interrupted. For mechanical assembly shops of machine-building production, the average number of operations of one technological process is quite large (15 - 80). If we consider this number as a base number and assume that, on average, in the manufacture of one assembly unit, a small set of enlarged types of work is used (turning, locksmith, milling, etc.),
then the resulting distribution can be successfully used to assess the impact of stochastic disturbances on the course of the production process.
The author conducted a simulation experiment built on this principle. To generate a sequence of pseudo-random variables uniformly distributed over the interval 0.9 - 1.0, a pseudo-random number generator was used, described in . The software of the experiment is written in the COBOL algorithmic language.
In the experiment, products of generated random variables are formed, simulating the real probabilities of the complete execution of a specific technological process. They are compared with the probability of performing the technological process, obtained using the geometric mean value, which was calculated for a certain sequence of random numbers of the same distribution. The geometric mean is raised to a power equal to the number of factors in the product. Between these two results, the relative difference in percent is calculated. The experiment is repeated for a different number of factors in the products and the number of numbers for which the geometric mean is calculated. A fragment of the results of the experiment is shown in Table 2.
table 2
Simulation experiment results:
n is the degree of the geometric mean; k - the degree of the product
n to Product Deviation to Product Deviation to Product Deviation
10 1 0,9680 0% 7 0,7200 3% 13 0,6277 -7%
10 19 0,4620 -1% 25 0,3577 -1% 31 0,2453 2%
10 37 0,2004 6% 43 0,1333 4% 49 0,0888 6%
10 55 0,0598 8% 61 0,0475 5% 67 0,0376 2%
10 73 0,0277 1% 79 0,0196 9% 85 0,0143 2%
10 91 0,0094 9% 97 0,0058 0%
13 7 0,7200 8% 13 0,6277 0% 19 0,4620 0%
13 25 0,3577 5% 31 0,2453 6% 37 0,2004 4%
13 43 0,1333 3% 49 0,0888 8% 55 0,0598 8%
13 61 0,0475 2% 67 0,0376 8% 73 0,0277 2%
13 79 0,0196 1% 85 0,0143 5% 91 0,0094 5%
16 1 0,9680 0% 7 0,7200 9%
16 13 0,6277 2% 19 0,4620 3% 25 0,3577 0%
16 31 0,2453 2% 37 0,2004 2% 43 0,1333 5%
16 49 0,0888 4% 55 0,0598 0% 61 0,0475 7%
16 67 0,0376 5% 73 0,0277 5% 79 0,0196 2%
16 85 0,0143 4% 91 0,0094 0% 97 0,0058 4%
19 4 0,8157 4% 10 0,6591 1% 16 0,5795 -9%
19 22 0,4373 -5% 28 0,2814 5% 34 0,2256 3%
19 40 0,1591 6% 46 0,1118 1% 52 0,0757 3%
19 58 0,0529 4% 64 0,0418 3% 70 0,0330 2%
19 76 0,0241 6% 82 0,0160 1% 88 0,0117 8%
19 94 0,0075 7% 100 0,0048 3%
22 10 0,6591 4% 16 0,5795 -4% 22 0,4373 0%
22 28 0,2814 5% 34 0,2256 5% 40 0,1591 1%
22 46 0,1118 1% 52 0,0757 0% 58 0,0529 8%
22 64 0,0418 1% 70 0,0330 3% 76 0,0241 5%
22 82 0,0160 4% 88 0,0117 2% 94 0,0075 5%
22 100 0,0048 1%
25 4 0,8157 3% 10 0,6591 0%
25 16 0,5795 0% 72 0,4373 -7% 28 0,2814 2%
25 34 0,2256 9% 40 0,1591 1% 46 0,1118 4%
25 52 0,0757 5% 58 0,0529 4% 64 0,0418 2%
25 70 0,0330 0% 76 0,0241 2% 82 0,0160 4%
28 4 0,8157 2% 10 0,6591 -2% 16 0,5795 -5%
28 22 0,4373 -3% 28 0,2814 2% 34 0,2256 -1%
28 40 0,1591 6% 46 0,1118 6% 52 0,0757 1%
28 58 0,0529 4% 64 0,041 8 9% 70 0,0330 5%
28 70 0,0241 2% 82 0,0160 3% 88 0,0117 1%
28 94 0,0075 100 0,0048 5%
31 10 0,6591 -3% 16 0,5795 -5% 22 0,4373 -4%
31 28 0,2814 0% 34 0,2256 -3% 40 0,1591 4%
31 46 0,1118 3% 52 0,0757 7% 58 0,0529 9%
31 64 0,0418 4% 70 0,0330 0% 76 0,0241 6%
31 82 0,0160 6% 88 0,0117 2% 94 0,0075 5%
When setting up this simulation experiment, the goal was to explore the possibility of obtaining, using the probability distribution (2), one of the enlarged statistical characteristics of the production process - the probability of performing one technological process of manufacturing an assembly unit consisting of K operations without failures. For a specific technological process, this probability is equal to the product of the probabilities of performing all its operations. As the simulation experiment shows, its relative deviations from the probability obtained using the developed probabilistic model do not exceed 9%.
Since the simulation experiment uses a more inconvenient than real probability distribution, the practical discrepancies will be even smaller. Deviations are observed both in the direction of decreasing and in the direction of exceeding the value obtained from the average characteristics. This fact suggests that if we consider the deviation of the probability of failure-free execution of not a single technological process, but several, then it will be much less. Obviously, it will be the smaller, the more technological processes will be considered. Thus, the simulation experiment shows a good agreement between the probability of performing without failures of the technological process of manufacturing products with the probability obtained using a one-parameter mathematical model.
In addition, simulation experiments were carried out:
To study the statistical convergence of the probability distribution parameter estimate;
To study the statistical stability of the mathematical expectation of the number of operations performed without failures;
To analyze methods for determining the duration of the minimum planning period and assessing the discrepancy between planned and actual indicators of the production process, if the planned and production periods do not coincide in time.
Experiments have shown good agreement between the theoretical data obtained through the use of techniques and the empirical data obtained by simulation on
Series "Economics and Management"
Computer of real production processes.
Based on the application of the constructed mathematical model, the author has developed three specific methods for improving the efficiency of operational management. For their approbation, separate simulation experiments were carried out.
1. Methodology for determining the rational volume of the production task for the planning period.
2. Methodology for determining the most effective duration of the operational planning period.
3. Evaluation of the discrepancy in the event of a mismatch in time between the planned and production periods.
Literature
1. Mordasov Yu.P. Determining the duration of the minimum operational planning period under the action of random disturbances / Economic-mathematical and simulation modeling using computers. - M: MIU im. S. Ordzhonikidze, 1984.
2. Naylor T. Machine simulation experiments with models of economic systems. -M: Mir, 1975.
The transition from concentration to diversification is an effective way to develop the economy of small and medium-sized businesses
prof. Kozlenko N. N. University of Mechanical Engineering
Annotation. This article considers the problem of choosing the most effective development of Russian small and medium-sized businesses through the transition from a concentration strategy to a diversification strategy. The issues of diversification feasibility, its advantages, criteria for choosing the path of diversification are considered, a classification of diversification strategies is given.
Key words: small and medium businesses; diversification; strategic fit; competitive advantages.
An active change in the parameters of the macro environment (changes in market conditions, the emergence of new competitors in related industries, an increase in the level of competition in general) often leads to non-fulfillment of the planned strategic plans of small and medium-sized businesses, loss of financial and economic stability of enterprises due to a significant gap between the objective conditions for the activities of small businesses. enterprises and the level of technology of their management.
The main conditions for economic stability and the possibility of maintaining competitive advantages are the ability of the management system to respond in a timely manner and change internal production processes (change the assortment taking into account diversification, rebuild production and technological processes, change the structure of the organization, use innovative marketing and management tools).
A study of the practice of Russian small and medium-sized enterprises of production type and service has revealed the following features and basic cause-and-effect relationships regarding the current trend in the transition of small enterprises from concentration to diversification.
Most SMBs start out as small, one-size-fits-all businesses serving local or regional markets. At the beginning of its activity, the product range of such a company is very limited, its capital base is weak, and its competitive position is vulnerable. Typically, the strategy of such companies focuses on sales growth and market share, as well as
As the name implies, this type of model is focused on the description of systems that exhibit statistically regular random behavior, and time in them can be considered as a discrete value. The essence of time discretization is the same as in discrete-deterministic models. Models of systems of this kind can be built on the basis of two formalized description schemes. First, these are finite-difference equations, among the variables of which are functions that define random processes. Secondly, they use probabilistic automata.
An example of constructing a discrete stochastic system. Let there be some production system, the structure of which is shown in Fig. 3.8. Within the framework of this system, a homogeneous material flow moves through the stages of storage and production.
Let, for example, the flow of raw materials consist of metal ingots, which are stored in the input warehouse. Then these discs go to production, where some kind of product is produced from them. Finished products are stored in the output warehouse, from where they are taken for further actions with them (transferred to the next phases of production or for sale). In the general case, such a production system converts the material flows of raw materials, materials and semi-finished products into a flow of finished products.
Let the time step in this production system be equal to one (D? = 1). We will take the change in the operation of this system as a unit. We assume that the manufacturing process of the product lasts one time step.
Rice. 3.8, Production system diagram
The production process is controlled by a special regulatory body, which is given a plan for the release of products in the form of a directive intensity of output (the number of products that must be manufactured per unit of time, in this case, per shift). We denote this intensity d t . In fact, this is the rate of production. Let be d t \u003d a + bt, i.e. is a linear function. This means that with each subsequent shift, the plan increases by bt.
Since we are dealing with a homogeneous material flow, we believe that, on average, the volume of raw materials entering the system per unit of time, the volume of production per unit of time, the volume of finished products leaving the system per unit of time should be equal to d t .
The input and output flows for the regulatory body are uncontrollable, their intensity (or speed - the number of blanks or products per unit of time, respectively, entering and leaving the system) must be equal to d t . However, discs may be lost during transportation, or some of them will be of poor quality, or for some reason more than necessary will arrive, etc. Therefore, we assume that the input flow has an intensity:
x t in \u003d d t +ξ t in,
where ξ 1 in is a uniformly distributed random variable from -15 to +15.
Approximately the same processes can occur with the output stream. Therefore, the output flow has the following intensity:
x t in s x \u003d d t +ξ t out,
where ξ t out is a normally distributed random variable with zero mathematical expectation and variance equal to 15.
We will assume that in the production process there are accidents associated with the absence of workers for work, breakdowns of machines, etc. These randomnesses are described by a normally distributed random variable with zero mathematical expectation and a variance equal to 15. Let us denote it by ξ t/ The production process lasts a unit of time, during which x t raw materials, then these raw materials are processed and transferred to the output warehouse in the same unit of time. The regulatory body receives information about the operation of the system in three possible ways (they are marked with numbers 1, 2, 3 in Fig. 3.8). We believe that these methods of obtaining information are mutually exclusive in the system for some reason.
Method 1. The regulatory body receives only information about the state of the input warehouse (for example, about a change in stocks in a warehouse or about a deviation in the volume of stocks from their standard level) and uses it to judge the speed of the production process (about the speed of withdrawal of raw materials from the warehouse):
1) ( u t in - u t-1 in )- change in the volume of stocks in the warehouse (u t in - the volume of raw materials in the input warehouse at the time t);
2) (ù- u t in) - deviation of the volume of raw materials in the input warehouse from the stock rate.
Way 2. Regulator receives information directly from production (x t - actual production intensity) and compares it with the directive intensity (dt-xt).
Method 3. The regulatory body receives information as in method 1, but from the output warehouse in the form ( u t out - u t-1 out )- or (u -u t out). He also judges the production process on the basis of indirect data - an increase or decrease in stocks of finished goods.
To maintain a given production rate d t , the regulatory body makes decisions y t ,(or (y t - y t - 1)), aimed at changing the actual output intensity x t . As a decision, the regulatory body informs the production of the intensity values with which to work, i.e. x t = y t . The second version of the control solution - (yt-yt-1), those. the regulator tells production how much to increase or decrease the intensity of production (x t -x t-1).
Depending on the method of obtaining information and the type of variable that describes the control action, the following quantities can influence decision making.
1. Decision base (the value that should be equal to the actual intensity of production if there were no deviations):
directive output intensity at the moment t(dt);
the rate of change in the directive intensity of output at the moment t(dt-dt-1).
2. Deviation amount:
deviation of actual output from directive (dt-xt);
deviation of the actual volume of output from the planned volume
Σ d τ - Σ x τ
change in the level of stocks at the input ( ( u t in - u t-1 in) or output
(u t out - u t-1 out) warehouses;
stock level deviation at the input (ù- u t input) or output ( u -u t out) warehouses from the standard level.
In general, the management decision made by the regulatory body consists of the following components:
Solution examples:
y t = d t +y(d t-1 -x t-1);
y t = d t -y(ù -u t out)
Taking various decisions in form, the regulatory body seeks to achieve the main goal - to bring the actual output intensity closer to the directive one. However, he cannot always be directly guided in his decisions by the degree to which this goal is achieved. (dt - xt). The final results can be expressed in the achievement of local goals - stabilization of the level of stocks in the input or output warehouse ( and t in (out) - and t-1 in (out)) or in the approximation of the level of stocks in the warehouse to the standard (and-and in (out)). Depending on the goal to be achieved, the type of sign (+ or -) in front of the mismatch fraction used for regulation is determined in the control solution.
Let in our case, the regulatory body receives information about the state of the input warehouse (change in the level of stocks). It is known that in any control system there are delays in the development and implementation of a solution. In this example, information about the state of the input warehouse enters the regulatory body with a delay of one time step. Such a delay is called a decision delay and means that by the time the information is received by the regulatory body, the actual state of the stock level in the input warehouse will already be different. Once the regulator has made a decision at t it will also take time (in our example it will be a unit of time) to bring the solution to the performer. This means that the actual intensity of production is not y t , but to the decision that the governing body made a unit of time ago. This is a delay in the implementation of the solution.
To describe our production system, we have the following equations:
x tbx=d t +ξ t in
x t exit =dt +ξ t out;
y t = dt + y(u -u t-2 in)
x t = y t-1 + ξt
u t in - u t-1 in = x t in - x t
This system of equations allows you to build a model of the production system, in which the input variables will be d t ,ξ t in, ξ t out, ξ t ,a
day off - x t . This is true because an external observer views our production as a system that receives raw materials at a rate dt and producing products with intensity x t , subject to randomness ξ t in, ξ t out, ξ t . Having carried out all the substitutions in the resulting system of equations, we arrive at one equation of dynamics that characterizes the behavior x t depending on the d t ,ξ t in, ξ t out, ξ t .
The model considered above did not contain restrictions on the volume of warehouses and production capacities. If we assume that the capacity of the input warehouse is Vx, the capacity of the output warehouse is V BX, and the production capacity is M, then the new system of equations for such a nonlinear production system will be as follows:
x tBX=min((d t+ ξ t in), (V in - u t in)) - it is impossible to put more into the input warehouse than space will allow;
x exit =min((d t+ ξ t out),(V out - u t out)) - you cannot take more products from the output warehouse than there are;
y t =d t + y(u t in -u t-1 in)
x tBX = min(( u t in, ( y t-1+ ξ t in), M,(V out - u t out)) - it is impossible to produce more products than ordered, the limiting factors are the number of blanks available and the availability of free space in the output warehouse;
u t in -u t-1 in = x tBX-x t
The construction of a stochastic model includes the development, quality assessment and study of the system behavior using equations that describe the process under study.
To do this, by conducting a special experiment with a real system, the initial information is obtained. In this case, methods of planning an experiment, processing results, as well as criteria for evaluating the obtained models, based on such sections of mathematical statistics as dispersion, correlation, regression analysis, etc., are used.
The methods for constructing a statistical model describing the technological process (Fig. 6.1) are based on the concept of a "black box". Multiple measurements of input factors are possible for it: x 1 ,x 2 ,…,x k and output parameters: y 1 ,y 2 ,…,y p, according to the results of which dependencies are established:
In statistical modeling, following the formulation of problem (1), the least important factors are screened out from a large number of input variables that affect the course of process (2). The input variables selected for further research make up a list of factors x 1 ,x 2 ,…,x k in (6.1), by controlling which it is possible to control the output parameters y n. The number of model outputs should also be reduced as much as possible to reduce the cost of experimentation and data processing.
When developing a statistical model, its structure (3) is usually set arbitrarily, in the form of convenient-to-use functions approximating experimental data, and then refined based on an assessment of the adequacy of the model.
The polynomial form of the model is most commonly used. So, for a quadratic function:
(6.2)
where b 0 , b i , b ij , b ii are the regression coefficients.
Usually, we first restrict ourselves to the simplest linear model, for which in (6.2) b ii =0, b ij =0. In case of its inadequacy, the model is complicated by the introduction of terms that take into account the interaction of factors x i ,x j and (or) quadratic terms .
In order to maximize the extraction of information from the ongoing experiments and reduce their number, experiments are planned (4) i.e. selection of the number and conditions for conducting experiments necessary and sufficient to solve the problem with a given accuracy.
To build statistical models, two types of experiments are used: passive and active. Passive experiment It is carried out in the form of long-term observation of the course of an uncontrolled process, which makes it possible to collect an extensive range of data for statistical analysis. AT active experiment it is possible to control the conditions of the experiments. When it is carried out, the most effective is the simultaneous variation of the magnitude of all factors according to a certain plan, which makes it possible to identify the interaction of factors and reduce the number of experiments.
Based on the results of the experiments (5), the regression coefficients (6.2) are calculated and their statistical significance is estimated, which completes the construction of the model (6). The measure of the adequacy of model (7) is the variance, i.e. standard deviation of the calculated values from the experimental ones. The obtained variance is compared with the admissible one with the achieved accuracy of the experiments.
480 rub. | 150 UAH | $7.5 ", MOUSEOFF, FGCOLOR, "#FFFFCC",BGCOLOR, "#393939");" onMouseOut="return nd();"> Thesis - 480 rubles, shipping 10 minutes 24 hours a day, seven days a week and holidays
Demidova Anastasia Vyacheslavovna Method for constructing stochastic models of one-step processes: dissertation ... Candidate of Physical and Mathematical Sciences: 05.13.18 / Demidova Anastasia Vyacheslavovna; [Place of defense: Peoples' Friendship University of Russia].- Moscow, 2014.- 126 p.
Introduction
Chapter 1. Review of works on the topic of the dissertation 14
1.1. Overview of population dynamics models 14
1.2. Stochastic population models 23
1.3. Stochastic Differential Equations 26
1.4. Information on stochastic calculus 32
Chapter 2 One-Step Process Modeling Method 39
2.1. One step processes. Kolmogorov-Chapman equation. Basic kinetic equation 39
2.2. Method for modeling multidimensional one-step processes. 47
2.3. Numerical simulation 56
Chapter 3 Application of the method of modeling one-step processes 60
3.1. Stochastic models of population dynamics 60
3.2. Stochastic models of population systems with various inter- and intraspecific interactions 75
3.3. Stochastic model of the spread of network worms. 92
3.4. Stochastic models of peer-to-peer protocols 97
Conclusion 113
Literature 116
Stochastic differential equations
One of the objectives of the dissertation is the task of writing a stochastic differential equation for a system so that the stochastic term is associated with the structure of the system under study. One possible solution to this problem is to obtain the stochastic and deterministic parts from the same equation. For these purposes, it is convenient to use the basic kinetic equation, which can be approximated by the Fokker-Planck equation, for which, in turn, one can write an equivalent stochastic differential equation in the form of the Langevin equation.
Section 1.4. contains the basic information necessary to indicate the relationship between the stochastic differential equation and the Fokker-Planck equation, as well as the basic concepts of stochastic calculus.
The second chapter provides basic information from the theory of random processes and, on the basis of this theory, a method for modeling one-step processes is formulated.
Section 2.1 provides basic information from the theory of random one-step processes.
One-step processes are understood as Markov processes with continuous time, taking values in the region of integers, the transition matrix of which allows only transitions between adjacent sections.
We consider a multidimensional one-step process Х() = (i(),2(), ...,n()) = ( j(), = 1, ) , (0.1) Є , where is the length of the time interval on which the X() process is specified. The set G \u003d (x, \u003d 1, Є NQ x NQ1 is the set of discrete values that a random process can take.
For this one-step process, the probabilities of transitions per unit time s+ and s from state Xj to state Xj__i and Xj_i, respectively, are introduced. In this case, it is considered that the probability of transition from state x to two or more steps per unit of time is very small. Therefore, we can say that the state vector Xj of the system changes in steps of length Г( and then instead of transitions from x to Xj+i and Xj_i, we can consider transitions from X to X + Гі and X - Гі, respectively.
When modeling systems in which temporal evolution occurs as a result of the interaction of system elements, it is convenient to describe using the main kinetic equation (another name is the master equation, and in the English literature it is called the Master equation).
Next, the question arises of how to obtain a description of the system under study, described by one-step processes, with the help of a stochastic differential equation in the form of the Langevin equation from the basic kinetic equation. Formally, only equations containing stochastic functions should be classified as stochastic equations. Thus, only the Langevin equations satisfy this definition. However, they are directly related to other equations, namely the Fokker-Planck equation and the basic kinetic equation. Therefore, it seems logical to consider all these equations together. Therefore, to solve this problem, it is proposed to approximate the main kinetic equation by the Fokker-Planck equation, for which it is possible to write an equivalent stochastic differential equation in the form of the Langevin equation.
Section 2.2 formulates a method for describing and stochastic modeling of systems described by multidimensional one-step processes.
In addition, it is shown that the coefficients for the Fokker-Planck equation can be obtained immediately after writing for the system under study the interaction scheme, the state change vector r and expressions for the transition probabilities s+ and s-, i.e. in the practical application of this method, there is no need to write down the main kinetic equation.
Section 2.3. the Runge-Kutta method for the numerical solution of stochastic differential equations is considered, which is used in the third chapter to illustrate the results obtained.
The third chapter presents an illustration of the application of the method of constructing stochastic models described in the second chapter, using the example of systems describing the dynamics of the growth of interacting populations, such as "predator-prey", symbiosis, competition and their modifications. The aim is to write them as stochastic differential equations and to investigate the effect of introducing stochastics on the behavior of the system.
In section 3.1. the application of the method described in the second chapter is illustrated on the example of the “predator-prey” model. Systems with the interaction of two types of populations of the “predator-prey” type have been widely studied, which makes it possible to compare the results obtained with those already well known.
The analysis of the obtained equations showed that to study the deterministic behavior of the system, one can use the drift vector A of the obtained stochastic differential equation, i.e. The developed method can be used to analyze both stochastic and deterministic behavior. In addition, it was concluded that stochastic models provide a more realistic description of the behavior of the system. In particular, for the “predator-prey” system in the deterministic case, the solutions of the equations have a periodic form and the phase volume is preserved, while the introduction of stochastics into the model gives a monotonous increase in the phase volume, which indicates the inevitable death of one or both populations. In order to visualize the results obtained, numerical simulation was carried out.
Section 3.2. The developed method is used to obtain and analyze various stochastic models of population dynamics, such as the "predator-prey" model, taking into account interspecific competition among prey, symbiosis, competition, and the model of the interaction of three populations.
Information on stochastic calculus
The development of the theory of random processes led to a transition in the study of natural phenomena from deterministic representations and models of population dynamics to probabilistic ones and, as a result, the emergence of a large number of works devoted to stochastic modeling in mathematical biology, chemistry, economics, etc.
When considering deterministic population models, such important points as the random influences of various factors on the evolution of the system remain uncovered. When describing population dynamics, one should take into account the random nature of reproduction and survival of individuals, as well as random fluctuations that occur in the environment over time and lead to random fluctuations in system parameters. Therefore, probabilistic mechanisms that reflect these moments should be introduced into any model of population dynamics.
Stochastic modeling allows a more complete description of changes in population characteristics, taking into account both all deterministic factors and random effects that can significantly change the conclusions from deterministic models. On the other hand, they can be used to reveal qualitatively new aspects of population behavior.
Stochastic models of changes in population states can be described using random processes. Under some assumptions, we can assume that the behavior of the population, given its present state, does not depend on how this state was achieved (i.e., with a fixed present, the future does not depend on the past). That. To model the processes of population dynamics, it is convenient to use Markov birth-death processes and the corresponding control equations, which are described in detail in the second part of the paper.
N. N. Kalinkin in his works to illustrate the processes occurring in systems with interacting elements uses interaction schemes and, on the basis of these schemes, builds models of these systems using the apparatus of branching Markov processes. The application of this approach is illustrated by the example of modeling processes in chemical, population, telecommunication, and other systems.
The paper considers probabilistic population models, for the construction of which the apparatus of birth-death processes is used, and the resulting systems of differential-difference equations are dynamic equations for random processes. The paper also considers methods for finding solutions to these equations.
You can find many articles devoted to the construction of stochastic models that take into account various factors that affect the dynamics of changes in population size. So, for example, in the articles a model of the dynamics of the size of a biological community is built and analyzed, in which individuals consume food resources containing harmful substances. And in the model of population evolution, the article takes into account the factor of settling of representatives of populations in their habitats. The model is a system of self-consistent Vlasov equations.
It is worth noting the works that are devoted to the theory of fluctuations and the application of stochastic methods in the natural sciences, such as physics, chemistry, biology, etc. birth-death processes.
One can consider the “predator-prey” model as a realization of birth-death processes. In this interpretation, they can be used for models in many fields of science. In the 1970s, M. Doi proposed a method for studying such models based on creation-annihilation operators (by analogy with second quantization). Here you can mark the work. In addition, this method is now being actively developed in the group of M. M. Gnatich.
Another approach to modeling and studying models of population dynamics is associated with the theory of optimal control. Here you can mark the work.
It can be noted that most of the works devoted to the construction of stochastic models of population processes use the apparatus of random processes to obtain differential-difference equations and subsequent numerical implementation. In addition, stochastic differential equations in the Langevin form are widely used, in which the stochastic term is added from general considerations about the behavior of the system and is intended to describe random environmental effects. Further study of the model is their qualitative analysis or finding solutions using numerical methods.
Stochastic differential equations Definition 1. A stochastic differential equation is a differential equation in which one or more terms represent a stochastic process. The most used and well-known example of a stochastic differential equation (SDE) is an equation with a term that describes white noise and can be viewed as a Wiener process Wt, t 0.
Stochastic differential equations are an important and widely used mathematical tool in the study and modeling of dynamic systems that are subject to various random perturbations.
The beginning of stochastic modeling of natural phenomena is considered to be the description of the phenomenon of Brownian motion, which was discovered by R. Brown in 1827, when he studied the movement of plant pollen in a liquid. The first rigorous explanation of this phenomenon was independently given by A. Einstein and M. Smoluchowski. It is worth noting the collection of articles in which the works of A. Einstein and M. Smoluchowski on Brownian motion are collected. These studies have made a significant contribution to the development of the theory of Brownian motion and its experimental verification. A. Einstein created a molecular kinetic theory for the quantitative description of Brownian motion. The obtained formulas were confirmed by the experiments of J. Perrin in 1908-1909.
Method for modeling multidimensional one-step processes.
To describe the evolution of systems with interacting elements, there are two approaches - this is the construction of deterministic or stochastic models. Unlike deterministic, stochastic models allow taking into account the probabilistic nature of the processes occurring in the systems under study, as well as the effects of the external environment that cause random fluctuations in the model parameters.
The subject of study are systems, the processes occurring in which can be described using one-step processes and those in which the transition of their one state to another is associated with the interaction of the elements of the system. An example is models that describe the growth dynamics of interacting populations, such as "predator-prey", symbiosis, competition and their modifications. The aim is to write down for such systems the SDE and to investigate the influence of the introduction of the stochastic part on the behavior of the solution of the equation describing the deterministic behavior.
Chemical kinetics
The systems of equations that arise when describing systems with interacting elements are in many ways similar to systems of differential equations that describe the kinetics of chemical reactions. Thus, for example, the Lotka-Volterra system was originally deduced by Lotka as a system describing some hypothetical chemical reaction, and only later Volterra deduced it as a system describing the "predator-prey" model.
Chemical kinetics describes chemical reactions with the help of the so-called stoichiometric equations - equations reflecting the quantitative ratios of the reactants and products of a chemical reaction and having the following general form: where the natural numbers mі and U are called stoichiometric coefficients. This is a symbolic record of a chemical reaction in which ti molecules of the reagent Xi, ni2 molecules of the reagent Xp, ..., tr molecules of the reagent Xp, having entered into the reaction, form u molecules of the substance Yї, u molecules of the substance I2, ..., nq molecules of the substance Yq, respectively .
In chemical kinetics, it is believed that a chemical reaction can occur only with the direct interaction of reagents, and the rate of a chemical reaction is defined as the number of particles formed per unit time per unit volume.
The basic postulate of chemical kinetics is the law of mass action, which says that the rate of a chemical reaction is directly proportional to the product of the concentrations of reactants in powers of their stoichiometric coefficients. Therefore, if we denote by XI and y I the concentrations of the corresponding substances, then we have an equation for the rate of change in the concentration of a substance over time as a result of a chemical reaction:
Further, it is proposed to use the basic ideas of chemical kinetics to describe systems whose evolution in time occurs as a result of the interaction of the elements of this system with each other, making the following main changes: 1. not the reaction rates are considered, but the transition probabilities; 2. it is proposed that the probability of a transition from one state to another, which is the result of an interaction, is proportional to the number of possible interactions of this type; 3. To describe the system in this method, the main kinetic equation is used; 4. deterministic equations are replaced by stochastic ones. A similar approach to the description of such systems can be found in the works. To describe the processes occurring in the simulated system, it is supposed to use, as noted above, Markov one-step processes.
Consider a system consisting of types of different elements that can interact with each other in various ways. Denote by an element of the -th type, where = 1, and by - the number of elements of the -th type.
Let be (), .
Let's assume that the file consists of one part. Thus, in one step of interaction between the new node that wants to download the file and the node that distributes the file, the new node downloads the entire file and becomes the distributor node.
Let is the designation of the new node, is the distributing node, and is the interaction coefficient. New nodes can enter the system with intensity, and distributing nodes can leave it with intensity. Then the interaction scheme and the vector r will look like:
A stochastic differential equation in the Langevin form can be obtained 100 using the corresponding formula (1.15). Because the drift vector A fully describes the deterministic behavior of the system, you can get a system of ordinary differential equations that describe the dynamics of the number of new customers and seeds:
Thus, depending on the choice of parameters, the singular point can have a different character. Thus, for /3A 4/I2, the singular point is a stable focus, and for the inverse relation, it is a stable node. In both cases, the singular point is stable, since the choice of coefficient values, changes in system variables can occur along one of two trajectories. If the singular point is a focus, then damped oscillations in the numbers of new and distributing nodes occur in the system (see Fig. 3.12). And in the nodal case, the approximation of numbers to stationary values occurs in a vibrationless mode (see Fig. 3.13). The phase portraits of the system for each of the two cases are shown, respectively, in graphs (3.14) and (3.15).
