Kolmogorov Differential Equations Research Articles

Time dependent performance analysis of a Smart Trash bin using state-based Markov model and Reliability approach

In this paper, the authors have developed a quantitative mathematical model to assess the performance of a smart trash bin. This model takes into account six hardware components of the trash bin, including Arduino Uno, ultrasonic sensor, servo motor, switch, battery, jumper wires utilizing Markov modelling and reliability-based approach. The objective of this model is to facilitate timely planning of repair and maintenance activities, ensuring prolonged availability of the smart trash bin after identifying the weakest component/components of the system.The efficient functioning of all components in the smart trash bin is imperative for the timely transmission of the garbage level data to the municipal corporation. To achieve this, the paper focuses on evaluating key reliability metric for smart trash bin system, specifically reliability, unreliability, and Mean time to Failure (MTTF). The modelling approach employs a state-based Markov model, from which Kolmogorov differential equations are derived and subsequently solved using Laplace transformation. After collecting data from organization and experts of the field explicit expression for system reliability, unreliability and MTTF have been obtained.Furthermore, this study includes sensitivity analysis of the system to pinpoint critical components which is switch and servo motor, requiring attention for proper maintenance. The research finding indicates that reliability and MTTF of the system diminish as the failure rate of the components increase under certain conditions.

Terminology, structure and model of the electrical systems operation process

The content of operation as one of the most important stages of the life cycle of a technical system is revealed. The structure of the process of technical operation of electrical engineering systems is considered. The essence of the individual stages of operation is revealed, their functional purpose is given. It is shown that some fundamental concepts related to operation do not have an unambiguous interpretation. The well-known definitions of these concepts are considered and a one-digit formulation of each of them, which corresponds to the current standards, is given. Special attention is paid to the disclosure of the content of the concept of state and consideration of various types of technical states of the system. The definition of the “condition” category, the formulation of which differs from the known definitions and allows more fully and accurately solving the problems of diagnosing the studied electrical systems, is given. An important feature of this definition is that it fixes the continuous variability of the state of any system in a function of time. At the same time, stable technical conditions that differ significantly from each other and correspond to the current version of the standard are distinguished. Such states are serviceable, faulty, operable, inoperable, partially inoperable, marginal. Functional conditions that are determined by the operating modes of the system are considered. The current definitions of the most important properties and events, among which a special place is occupied by reliability, survivability, safety, failure, damage, defect, are given. For the selected technical states, a model in the form of an oriented graph is synthesized. It is shown that the actual process of an electrical system operation has two features. The first one is that the probability of transition from one state to another does not depend on the type of state the system is in at the moment. The second feature is that the time spent in each technical condition can be either deterministic or random. The conclusion about the possibility of modeling the process of changing the technical states of the system by semi- Markov models with a discrete set of states is made. A mathematical model in the form of Kolmogorov differential equations, which makes it possible to estimate the probabilities of the system being in a particular state, has been compiled.

An approach to managing the stability parameters of complex infocommunication systems under destructive influences

The paper considers the approach to managing the stability parameters of complex infocommunication systems based on the structural-parametric method of analysis and synthesis of the state of opposing systems. An algorithm for controlling the parameters of the system's resistance to destructive influences with a priori uncertainty of the intentions of the impact is proposed. The solution of the problem of increasing the stability of the infocommunication system in the dynamics of the development of an antagonistic conflict using a system of Kolmogorov differential equations and the optimal distribution of its protective resource is obtained. The proposed approach, the developed algorithm and the results obtained can be used to justify the requirements for stability parameters and their implementation in infocommunication systems as part of the transition to methods of increasing its stability based on artificial intelligence.

Intelligent Management of Enterprise Business Processes

The article develops the conceptual foundations of natural and artificial intellectualization of the enterprise, as well as the combination of artificial and natural intelligence in managing business processes in the context of modern challenges of the business environment. Based on the methods of structural design, a system model of the enterprise is developed as the basis for intelligent management. Quantitative and qualitative effects of human–cyber–physical systems, which are the result of management intellectualization, are highlighted. The possibilities of using deviation and perturbation management methods in managing the state of enterprise development with the support of decision-making and implementation of an intelligent information system are considered. The features of making managerial decisions during intelligent enterprise management are considered. The place of the human factor in such intellectual management is highlighted, in particular, in terms of improving the intelligence of employees and the natural intellectualization of the enterprise. The problem of assessing and forecasting the state of enterprise development in the context of intellectual management is highlighted. In this context, the expediency of using mathematical methods of Markov process theory, using systems of Kolmogorov differential equations and their solutions, using numerical methods and applied software products is justified. This made it possible to study the dynamics of probabilities of states and stability of development of enterprises and their employees; the dynamics of probabilities of states of innovative and technological processes; scientific and technological, environmental, social and economic efficiency of a business. To test the proposed mathematical models for assessing and predicting the state of development of enterprises and their employees, appropriate studies were conducted on the sanatorium–resort complex in Truskavets (Ukraine).

ANALYSIS OF MARСOVIAN SYSTEMS WITH A GIVEN SET OF SELECTED STATES

Analysis of stationary Marcovian systems is traditionally performed using systems of linear Kolmogorov differential equations. Such systems make it possible to determine the probability of the analyzed system being in each of its possible states at an arbitrary time. This standard task becomes more complicated if the set of possible states of systems is heterogeneous and some special subset can be distinguished from it, in accordance with the specifics of the system functioning. Subject of the study is technology development for such systems analysis. In accordance with this, the purpose of the work is to find the distribution law of the random duration of such a system's stay on a set of possible states until it falls into a selected subset of these states. Method for solving the problem is proposed based on splitting the entire set of possible states of the system into two subsets. The first of them contains a selected subset of states, and the second contains all the other states of the system. Now a subset of states is allocated from the second subset, from which a direct transition to the states of the first subset is possible. Next, a system of differential equations describing the transitions between the formed subsets is formed. The solution of this system of equations gives the desired result – distribution of the random duration of the system's stay until the moment of the first hit in the selected subset of states. The method allows solving a large number of practical problems, for example, in the theory of complex systems reliability with many different failure states. In particular, finding the law of the uptime duration distribution, calculating the average duration of uptime.

A Multistate Markov Model for the Progression of Congestive Heart Failure Using Kolmogorov Differential Equation

The reliability of individuals and human activities are vital for the new complexity of life, work and medication improvement. Markov models are grounded strategies for assessing paces of progress between phases of chronic illnesses. This paper is mainly focused on the survival analysis of a heart patient with Congestive cardiac failure. The objective of the paper is to describe the progression process of chronic heart failure (CHF), to estimate the life expectancy of patient suffering from congestive Heart failure. Explicit expression for Availability is also derived in this paper by using Kolmogorov differential equations. Graphs have been plotted to determine the behaviour of life expectancy of patient and availability analysis.

THE STATES’ FINAL PROBABILITIES ANALYTICAL DESCRIPTION IN AN INCOMPLETELY ACCESSIBLE QUEUING SYSTEM WITH REFUSALS

Context. There is a problem of forecasting the efficiency of real queuing systems with refusals in the case of incomplete accessibility of service devices for the input flow of requirements. The solution of problem is necessary to create the possibility of more accurate design and control of such systems operation in real time. Objective. The aim of the research is to obtain an analytical description of the state’s final probabilities in a Markov queuing system with refusals and with incomplete accessibility of service devices for the input flow of requirements that is necessary to forecast the values of the queuing system performance indicators. Method. The probabilities of queuing systems’ states with refusals in the case of incomplete accessibility of service devices for the input flow of requirements are described by Kolmogorov differential equations. In a stationary state, these equations are transformed into a linearly dependent homogeneous system of algebraic equations. The number of equations is determined by the setdegree and for modern queuing and communication systems can be in the thousands, millions and more. Therefore, an attempt to predict the efficiency of a system is faced with the need to write down and numerically solve a countable set of algebraic equations systems that is quite difficult. The key idea of the proposed method for finding an analytical description of final probabilities for a given queuing system was the desire to move from the description of individual states (of 2n amount) to the description of groups of system states (of n+1 number) and to localize the influence of incomplete accessibility of service devices for the input flow of requirements in multiplicative functions of incomplete accessibility. Such functions allow obtaining the required analytical description and assessing the degree of the final probabilities transformation, in comparison with known systems, as well as assessing the forecasted values of the noted queuing system’s efficiency indicators when building a system and choosing the parameters for its controlling. Results. For the first time analytical expressions are obtained for the final probabilities of the queuing system states with refusals and with incomplete accessibility of service devices for the input flow of requirements, which makes it possible to evaluate as well as forecast values of all known system efficiency indicators. Conclusions. The resulting description turned out to be a general case for well-known type of Markov queuing systems with refusals. The results of the numerical experiment testify in favor of correctness the obtained analytical expressions for the final probabilities and in favor of possibility for their practical application in real queuing systems when solving problems of forecasting efficiency, as well as analyzing and synthesizing the parameters of real queuing systems.

MANAGEMENT IN HIGH-DIMENSIONAL MARKOV SYSTEMS

A problem of analyzing Markov systems along with a large number of states has been considered. The conventional computational procedure for obtaining analytical ratios for calculating the distribution of system states is based on the use of a system of Kolmogorov differential equations. The system of linear algebraic equations being formed later can be easily solved numerically. However, the complexity of obtaining an analytical solution increases rapidly with the increase in the problem dimension. In this regard, the purpose is to develop an effective method for studying Markov systems, the computational procedure of which ensures the possibility of obtaining solutions for high-dimensional models. The method is based on the decomposition of states graph and system transitions. The obtained analytical expressions allow to set and solve the problem of rational resource distribution for changing the values of the system parameters to increase its efficiency. The method ensures the possibility of solving management problems in Markov systems along with a large number of possible states. An example of method application has been considered.

Availability optimization of biological and chemical processing unit using genetic algorithm and particle swarm optimization

PurposeThe demand of sewage treatment plants is increasing day by day, especially in the countries like India. Biological and chemical unit of such sewage treatment plants are critical and needs to be designed and developed to achieve desired level of reliability, maintainability and availability.Design/methodology/approachThis paper investigates and optimizes the availability of biological and chemical unit of a sewage treatment plant. A novel mathematical model for this unit is developed using the Markovian birth-death process. A set of Chapman–Kolmogorov differential equations are derived for the model and a generalized solution is discovered using soft computing techniques namely genetic algorithm (GA) and particle swarm optimization (PSO).FindingsNature-inspired optimization techniques results of availability function depicted that PSO outperforms GA. The optimum value of the availability of biological and chemical processing unit is 0.9324 corresponding to population size 100, the number of evolutions 300, mutation 0.6 and crossover 0.85 achieved using GA while PSO results reflect that optimum achieved availability is 0.936240 after 45 iterations. Finally, it is revealed that PSO outperforms than GA.Research limitations/implicationsThis paper investigates and optimizes the availability of biological and chemical units of a sewage treatment plant. A novel mathematical model for this unit is developed using the Markovian birth-death process.Originality/valueAvailability model of biological and chemical units of a sewage treatment is developed using field failure data and judgments collected from the experts. Furthermore, availability of the system has been optimized to achieve desired level of reliability and maintainability.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ОРГАНИЗАЦИОННЫХ СИСТЕМ

Introduction. Modern rates of development of information systems and web-technologies are increasingly being introduced into the life of all mankind, including organizational systems, which requires the improvement of management and decision-making processes. In modern sources of literature, the consideration of management processes is given on the basis of analysis, but from the side of the synthesis of the mathematical model of management decisions, they are not considered, which allows us to consider the ongoing research as an urgent task. For the head of the organizational system, the solution of the problem of management is the most important task based on the complex interaction of all departments of the managed system. Practical relevance: When a destructive impact occurs in a managed system, a decision-maker should have a management decision model that allows timely management decisions to counter emerging threats. Counteraction is carried out at the expense of the resources available to the decision-maker. The mathematical model of a management decision makes it possible to respond in a timely manner to emerging threats in the system and make appropriate management decisions with a given level of the efficiency indicator of the decision being made. The application of the Kolmogorov differential equations makes it possible to link the finding of a mathematical modeling system with various states of the controlled system. Discussion: The resulting indicator of the effectiveness of the decision makes it possible to link three states of the system when considering the target process, namely, the process of threat formation, the process of threat identification and the process of threat neutralization. By setting the required indicator of the effectiveness of a management decision, it allows the manager to respond in a timely manner to the destructive impact, to use the available resources to ensure the achievement of the management goal.

On the Zero-Neutron Density in Stochastic Nuclear Dynamics

In this short paper, we compare the deterministic model for the nuclear reactor dynamic (Hetrick, 1993) with the stochastic model (Kinard and Allen, 2004). Our numerical results show coincidences between the deterministic model and the mean of the stochastic paths, although, as already observed by other authors, there is alarge amount of dispersion between the individual paths. Notably, we always observe that the neutron density approaches zero within a short time. In this paper, we investigate this question; more concretely, we study the mean-extinction of the neutron density. The technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using the finite-element method with FreeFem++. Our results confirm that in a very short time the neutrons disappear although later they recover probably due to the external source.

Assessment reliability parameters of the DVB-T2 broadcasting station’s equipment with local content insertion

Local content modification provided to subscribers of the terrestrial digital television signal is necessary to provide the technical possibility of organizing inserts of local content signals, such as TV programs of regional broadcasters, as well as information on emergency situations. Broadcast multiplexes of federal TV programs are designed for use within the corresponding time zone (A, B, C, D, M). In each time zone, there are a number of regions, in each of which there are local TV and radio companies that produce local content that must be delivered to the subscribers of the whole subject. The task of embedding/modifying content at each remote transmitting station is performed by an inserter or local content insertion device (ETSI TS 102773). The reliability parameters of the restorable system for organizing terrestrial television broadcasting at a remote station with the content modification were calculated in this article. Tables and a graph of the broadcasting system states are presented, on the basis of which, systems of Kolmogorov differential equations are compiled. It was found that additional redundancy organized by connecting the output stream from the RX1 receiver directly to the transmitting device allows for a 2.5-fold increase in the average operating time between failures, as well as an increase in the availability factor by 5.26 percent. All calculations were performed using the SimInTech software package. The influence of automatic redundancy of the local content inserter and the transmitter on the occurrence of errors in the stream that affect the quality of the output signal is considered. The relationship between the availability factor and the components of the quality of service parameter – SAE, SDE and SIE is determined.

Probabilistic model of penetration of biological agents through filtering devices

The human protection issue from biological agents with the help of filtering devices is studied in this article. This issue is actual for all the countries of the world due to the current coronavirus situation. Considering the hypothesis of the ordinariness of the biological agents’ flow and the absence of aftereffects in the filter, the probabilistic model of the penetration of biological agents through filtering respiratory protection devices is designed based on the Kolmogorov differential equation system applied for the adequate protection of medical personnel.

Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1.

Measurement and improvement of efficiency of Library by mathematical modelling and finding reliability

Here in this paper we have developed the mathematical model of the library server using Markov birth – death process assuming that library system server system is based on exponential distribution. The model so developed by victimisation Chapman Kolmogorov differential equation and is solved by using Mathematica. The solution so obtained is analysed for various rates of failures and repair. The finding so obtained are discussed with the concerned authorities of the library to boost the efficiency of the library.

Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics.

Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp.variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ В СОЦИАЛЬНЫХ И ЭКОНОМИЧЕСКИХ СИСТЕМАХ

In the modern rapidly changing world, there are constant changes in all spheres of human activity, including in the social and economic systems, such as education, law, health care. All areas require study for improving management and decision-making mechanisms in order to increase the efficiency of their functioning and further logic of action improving the process of functioning of processes in social and economic systems. The introduction of the sites of educational organizations into educational institutions of higher education (UHE) has led to the need for the person in charge of the site (LOU) of the organization to have a mathematical model of management decisions to counter emerging threats. In this paper, we will consider the process of forming a mathematical model of an administrative decision, which is obtained on the basis of synthesis from the transition states of the system using the Kolmogorov differential equations, by further transforming them into a system of linear algebraic equations (SLAE) and solving them by the Gauss method. Mathematical modeling is based on synthesis processes using the law of preserving the integrity of the object (ZSCO) and the natural science approach (ESA). The use of the synthesis method allows you to achieve the control goal based on the required performance indicators. The proposed mathematical model helps in solving three processes aimed at monitoring the occurrence of a problem in a controlled system, the process of recognizing the problem and the process of implementing a managerial decision to eliminate the problem. The final obtained mathematical solution for modeling the situation in the social and economic system helps to establish a model of LOU behavior depending on the current situation, which will lead to saving time resources and the possibility of its redistribution for solving other problems. The resulting mathematical model can be further complicated by adding new variables and conditions for the implementation of the control process.

A Comparison of Deterministic and Stochastic Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) Models

In this paper we build and analyze two stochastic epidemic models with death. The model assumes that only susceptible individuals (S) can get infected (I) and may die from this disease or a recovered individual becomes susceptible again (SIS model) or completely immune (SIR Model) for the remainder of the study period. Moreover, it is assumed there are no births, deaths, immigration or emigration during the study period; the community is said to be closed. In these infection disease models, there are two central questions: first it is the disease extinction or not and the second studies the time elapsed for such extinction, this paper will deal with this second question because the first answer corresponds to the basic reproduction number defined in the bibliography. More concretely, we study the mean-extinction of the diseases and the technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using finite element method with FreeFem++. Our contribution and novelty are the following: however the reproduction number effectively concludes the extinction or not of the disease, it does not help to know its extinction times because example with the same reproduction numbers has very different time. Moreover, the SIS model is slower, a result that is not surprising, but this difference seems to increase in the stochastic models with respect to the deterministic ones, it is reasonable to assume some uncertainly.

THE STATES FINAL PROBABILITIES ANALYTICAL DESCRIPTION IN QUEUING SYSTEM WITH AN ENTRANCE FLOW OF REQUIREMENTS GROUPS, WITH WAITING AND LEAVING THE QUEUE

Context. The problem of predicting the efficiency of real queuing systems in the event of a possible arrival of requirements groups and leaving of “impatient” requirements from the queue. The aim of the study was to model the operation of such systems to create opportunities to control their operation in real time. Objective. The aim of the research is to obtain an analytical description of the state’s final probabilities in a Markov queuing system with an input flow of requirements groups, with individual service of requirements, with a limited number of waiting places and with individual leaving of “impatient” requirements from the queue that is necessary to predict the values of the queuing system performance indicators.Method. The probabilities of queuing systems states with an input flow of requirements groups with a random composition and with leaving of “impatient” requirements from the queue are described by the Kolmogorov differential equations. In a stationary state, these equations are transformed into a linearly dependent homogeneous system of algebraic equations. The structure of the equations depends on the numerical values of the input flow requirements group’s parameters and the controlled service system. Therefore, an attempt to predict the efficiency of a system is faced with the need to write down and numerically solve a countable set of algebraic equations systems that is quite difficult. The key idea of the proposed method for finding an analytical description of the final probabilities for the specified queuing system was the desire to localize the influence of requirements groups in the input flow on the operation of the queuing system in multiplicative non-ordinary functions. Such functions allow obtaining the required analytical description and assessing the degree of the final probabilities transformation, in comparison with known systems, as well as assessing the predicted values of the noted queuing system efficiency indicators when choosing the parameters for controlling its operation. Results. For the first time analytical expressions are obtained for the final probabilities of the queuing system states with an input flow of random composition requirements groups, with a limited number of waiting places, with individual service and leaving “impatient” requirements from the queue, which makes it possible to evaluate all known indicators of the system’s performance. Conclusions. The resulting description turned out to be a general case for well-known types of Markov queuing systems with non-ordinary and with the simplest input flow of requirements. The results of the numerical experiment testify in favor of the correctness of the obtained analytical expressions for the final probabilities and in favor of the possibility of their practical application in real queuing systems when solving problems of forecasting efficiency, as well as analyzing and synthesizing the parameters of real queuing systems.

Performance improvement of a crystallization system through optimization and sensitivity analysis

PurposeThis paper deals with the performance optimization and sensitivity analysis for crystallization system of a sugar plant.Design/methodology/approachCrystallization system comprises of five subsystems, namely crystallizer, centrifugal pump and sugar grader. The Chapman–Kolmogorov differential equations are derived from the transition diagram of the crystallization system using mnemonic rule. These equations are solved to compute reliability and steady state availability by putting the appropriate combinations of failure and repair rates using normalizing and initial boundary conditions. The performance optimization is carried out by varying number of generations, population size, crossover and mutation probabilities. Finally, sensitivity analysis is performed to analyze the effect of change in failure rates of each subsystem on availability, mean time to failure (MTBF) and mean time to repair (MTTR).FindingsThe highest performance observed is 96.95% at crossover probability of 0.3 and sugar grader subsystem comes out to be the most critical and sensitive subsystem.Originality/valueThe findings of the paper highlights the optimum value of performance level at failure and repair rates for subsystems and also helps identify the most sensitive subsystem. These findings are highly beneficial for the maintenance personnel of the plant to plan the maintenance strategies accordingly.