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 Âûïóñê 77
- Sobolev V. Khinchin's basic law of a stationary queue for single-server queueing systems with batch arrivals
This paper deals with a queuing system with general renewal arrivals, exponential service times, single service channel and infinite number of waiting positions, customers are serviced in the order of their arrival. For this queueing system, a condition for the fulfilment of the Khinchin's basic law of a stationary queue is given. The article shows that, in the case of basic law of a stationary queue for our system, the stationary distribution of the number of the customers in the system always coincides with the corresponding probability distribution of the queueing system with exponential interarrival times. In stationary case a new form of the probability generating functions of the number of clients in the system is also derived. This new form is written in terms of the probability generating functions of the tail distribution function of the number of customers per group and of the probability generating functions of a embedded discrete time homogeneous Markov chain.
- Gorelov M. Hierarchical games with restrictions on content and volume of information transfered
Two players hierarchical game is investigated. The top-level player is supposed to have right to first move. It is supposed that he has access to some information about his partner’s choice. But two types of restrictions on such information are taken into consideration. From one hand there are such pairs of bottom-level player choices that elements of pair are not distinguished one from another from the top level-player’s point of view. From other hand the volume of information on the bottom-level player’s choice which the top-level player can handle is restricted. The combinatorial approach (in terms of A.N. Kolmogorov) is used for measuring of the amount of information. Top level player is supposed to have the right of choice of the “sense” of information obtained (in the framework of restrictions of the first type). It is assumed that the top-level player knows the opportunity and goals of his partner and he can expect to rational behavior of his partner. In such assumptions the problem of calculating of the top level player’s maximal guaranteed result is a problem of calculating a maxima on complex functional spaces. In the article the problem is reduced to calculation of multiple maximin on “finite-dimensional” spaces. Two approaches to computing of this result are proposed. A structure of top level player’s optimal strategy is estimated. In particular the optimal semantics of information which top laver player handle is estimated. An illustrative example is provided which demonstrates the possibilities of use of methods proposed.
- Furtat I., Gushchin P. Control algorithm under saturation of control signal and its derivatives
The synthesis of the control system under conditions of parametric uncertainties, bounded external disturbances, saturation of the control signal and its time derivatives is given. Such tasks are typical in the control of various kinds of technical systems. Ignoring uncertainties and constraints can lead to objective failure or loss of control system stability. For example, catastrophic consequences may occur in an aircraft’s flight control system due to the saturation of the controlled signal and its derivative. Other examples are the control of jet engine compressors and the study of the behavior of the control plant with a slow drive. To solve the problem in the conditions of uncertainties and restrictions on the regulating signal a new method of design the control law is proposed, which allows to ensure the finding of the regulating signal and its derivatives in given sets. The principle of compensation of disturbances is used, which allowed to obtain information about parametric uncertainty and external disturbances in the form of some function. To compensate these uncertainties a control signal is designed, the value of which is opposite to the value of perturbations. Thus, for control in the conditions of constraints on the control signal it is required to impose the corresponding constraints on the function containing information on these uncertainties. The conditions on the parameters of the plant, external disturbance, the reference model and the controller when performing the control system is feasible are obtained. The simulation results illustrate the performance of the proposed control scheme.
- Kvinto Y., Khlebnikov M. Upper bounds of large deviations~in linear discrete-time systems: the robust statement
The paper is devoted to the study of the important effect of large deviations in linear dynamical systems with nonzero initial conditions. The study of transients is actual and practically significant direction in the linear systems theory. The common Lyapunov quadratic function for the family of systems with uncertainties and the invariant ellipsoids approach are used in the article as main technical tools. All the results obtained are also applicablefor non-stationary uncertainties: the only condition for an uncertainty is its spectral norm constraint. The analysis and design problems are considered, and the upper bounds of deviations for linear discrete-time systems with structured matrix uncertainties are obtained. The obtained results have the form of semi-definite programs, which are easy to solve numerically via standart software packages. Using the technique of linear matrix inequalities, the problem of minimization the magnitude of deviations while stabilizing the system via the linear static state feedback was investigated. Numerical simulations demonstrate the low degree of conservatism of the obtained approach. The results have a great potential for generalizations.
- Belov M. Statistics of hiring and attrition processes
The paper presents research in the field of mathematical modeling of the economy of human capital and the life cycle of employees, as well as studying the dependency of the economic effects of business on these factors. The traffic of employees between the firm and the labor market is studied. The characteristics of the life cycle process of the employee (LCE), considered as the total time from the moment when the employee accepted the job offer from the company until he leaves the company. The materiality of the influence of the duration of the average work time in the firm, the duration of the employee's adaptation phase in hiring and the phase preceding the dismissal on the firm's economy is demonstrated. The process of dismissal of employees is analyzed by non-parametric methods. As a result, the impossibility of a highly reliable assessment of the probability distribution function of the employee’s time in the firm based on observations was shown. The procedures of statistic sequential analysis are proposed to identify the moment of change in the characteristics of the life cycle of employees - the probability of accepting job offers and the intensity of dismissal at will. The developed methods and the results obtained are illustrated by numerical examples from the author’s practice in the field of human capital economics and personnel management.
- Podvalny S., Vasiljev E. Multi-alternativity: the evolutionary strategy of biological systems
The article describes the results of mathematical modeling and quantitative analysis of the biological properties of diversity. The aim of the work is a methodological generalization of this property of living nature in the form of a cybernetic concept of multi-alternativeity as an evolutionary strategy for the functioning of complex systems. The study of the pre-biological stage of life origin, conducted on quasi-species and sisers models, showed that for the stable existence of macromolecules capable of self-replication, they must have cross mutational flows generating a variety of competing types of molecules. Analysis of the process of molecular genetic evolution at the level of populations was carried out on the model of the Markov random process with discrete states and allowed to quantitatively substantiate the condition of intra-species evolution: the growth rate of the genetic diversity of the population must exceed the rate of occurrence of environmental conditions that threaten its existence. To model evolutionary processes in a complex multi-level ecosystem with a closed trophic cycle, differential equations of mass balance were used for each trophic level of the hierarchy, taking into account the exchange of energy with the external environment as a mass of an equivalent substance. The results of ecosystem modeling showed that when a critical event occurs in it, the diversity of species due to the redistribution of their biomass again brings the system to the trajectory of balanced growth, and the presence of a hierarchical structure as a result of the influence of interlayer feedbacks significantly speeds up this process and makes it monotonous. The results of the work formulate a number of substantive principles that make up the concept of multi-alternative: the principles of the many-leveledness, separation of functions and discreteness. The constructive nature of these principles opens the possibility of practical implementation of the multi-alternative strategy in the construction of complex control systems for various purposes.
- Belov M., Novikov D. Models of designing and learning a technology of complex activity
Presented mathematical models are based on previous studies of the problems of managing organizational and technical systems and their complex activities, executed by the authors. The problem of developing and / or mastering the technology of complex activity is formalized in the form of a mathematical model, which generalizes of probabilistic learning models. The properties of the process of developing and / or mastering technology (learning) are studied, the convergence of the process to the state of full technology mastering is shown, analytical expressions of the characteristics of the models are obtained - the average time to reach a given level of mastering. The models of learning that describe the integration of elements of technology - conjunctive, disjunctive and parallel development of technology are proposed. Developed and studied models of learning in the process of work and group learning. including the “learning to learn” model - when the intensity of the learning process depends on the learning level achieved. For all models of integration, analytical expressions for learning levels are obtained. The asymptotic case of models during the transition to continuous time is investigated. It is shown that special cases of the proposed model are models of exponential, hyperbolic, and logistic learning curves, which are widespread in the theory of learning, theory of system testing, software testing, and related branches of knowledge.
- Guzha E., Romanenko V., Skorokhod M. Fuzzy assessment of performance of air transportation transfer system
The transfer system of passenger air transportation as part of the hub airport and the hub-forming airline is considered. The efficiency of the system is characterized by the level of its revenues from the transportation of transfer passengers with an optimally formed timetable. It is assumed that uncomfortable waiting times for a transfer can cause an outflow of passengers potentially ready to be transported through the hub formed by flights of certain airline, while minimizing the damage caused by the outflow of passengers allows optimization of the hub's timetable. The proposed model for assessing the effectiveness of the projected transfer system includes, as a key component, the timetable optimization task, which involves determining such moments of arrival and departure of flights in the hub that maximize revenues from transportation of transfer passengers taking into account capacity constrained of airport production complexes. Uncertainty in the initial data of the model, characteristic for the design stage of the transfer system, is presented in a fuzzy set form. The model can be used to support decision making on the economic efficiency and technical capacity of forming a transfer system based on certain airlines and airports, for solving timetable optimization tasks and creating a balanced fleet of technological resources at the hub airport, rational organization of its industrial complexes, operational resource management. Some examples of application of the proposed model are considered.
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