System Analysis of Educational Process Antonina V. Ganicheva 1 and Aleksey V. Ganichev 2 1 Tver State Agriculture Academy, 7 Marshall Vasilevskogo Street (Sakharov), Tver, 170026, Russia 2 Tver State Technical University, 22 Af. Nikitina Embankment, Tver, 170026, Russia Abstract One of the most important issues of the social sphere is the effective management of complex educational systems. The relevance and importance of solving this problem are determined by the modernization of Russian education, the process of its digitalization, and the introduction of new innovative technologies. Various methods and models are used for the mathematical description of educational organizations and the study of their functioning. None of these methods separately can solve all the problems in this area completely successfully. Therefore, the most convenient and effective method to solve them is system analysis. This article considers the application of Markovian chains for the system analysis of the educational process. The structural scheme of the educational process is developed and the educational system processes are described using discrete and continuous Markovian chains and network modeling. A method for analyzing the stability of the system of Kolmogorov equations, which corresponds to the considered model of the educational process, is formulated. The study resulted in the developed mathematical model of the educational system based on the system analysis method. The method proposed in the article can be used not only for managing educational organizations but also in other areas, for example, in economics and agriculture Keywords 1 Markovian chain, model, the system of differential equations 1. Introduction The administration of effective management of educational organizations is one of the key issues in the educational process [12]. This issue is becoming particularly relevant in our country in the context of Russian education modernization. Various methods and models are used to describe various processes in educational institutions. The main research area is the generalization of complex [8], system, process, competence [10] approaches, as well as the use of modern innovative technologies [5], including intelligent databases and knowledge [9]. For complex educational systems, the tool techniques of the Markovian chain theory are often used. In the articles [1, 3, 6, 7, 11] this mathematical tool technique is used to assess the indicators of the functioning of an educational institution and the creation of an educational environment. The article [4] takes up the use of Markovian chains for managing student learning. The use of Markovian processes for the study of the educational process should be considered from the system analysis perspective. In our opinion, insufficient attention has been paid to this topical issue in scientific research. The purpose of this article is the application of Markovian chains for the system analysis of the educational process. To achieve it, you need to address the following challenges: 1. To develop a structural scheme of the educational process. 2. To describe the processes in the system using discrete and continuous Markovian chains. 3. To examine the stability of the educational system. Proceedings of VI International Scientific and Practical Conference Distance Learning Technologies (DLT–2021), September 20-22, 2021, Yalta, Crimea EMAIL: TGAN55@yandex.ru (A. 1); alexej.ganichev@yandex.ru (A. 2) ORCID: 0000-0002-0224-8945 (A. 1); 0000-0003-3389-7582 (A. 2) ©️ 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) 161 2. Scheme of the Educational Process System The educational process takes place in a complex social system. The overall scheme of such a system is shown in Figure 1. Figure 1: Structural scheme of the educational process The following key shall apply for the scheme: Si - system status; Pi -probability of finding the system in Si ; Pij - probabilities of transition from Si to S j ; ti (i  0, 4) - system levels; EMC - educational and methodological complex. The numbering of the levels is made from the zero (initial) one. Transition probabilities Pij have the meaning of a fraction of efforts aimed at solving problems of this structural level. 3. Discrete Markovian Chain The conditions under which the learning process can be considered as Markovian are specified in the work [2]. For fixed stages, we ti have a discrete Markovian chain, the matrix of transition probabilities of which is as follows: 162  P01 P02 0 0 0 0 0 0   0 0 P13 P14 P15 0 0 0     0 0 0 0 P25 P26 0 0   0 0 0 0 0 0 P37 0 ,  0 0 0 0 0 0 P47 0     0 0 0 0 0 0 P57 0   0 0 0 0 0 0 P67 0   0 P78   0 0 0 0 0 0 provided that P0  1, P1 (t1 )  P01 , P2 (t1 )  P02 . (1) The remaining probabilities of the states are obtained by applying the formula of total probability:  P3 (t2 )  P13  P1 (t1 )  P13  P01 ; P4 (t2 )  P14  P1 (t1 )  P14  P01 ;  P5 (t2 )  P15  P1 (t1 )  P25  P2 (t1 )  P15  P01  P25  P02 ;   P6 (t2 )  P26  P2 (t1 )  P26  P02 ;  P7 (t3 )  P37  P3 (t2 )  P47  P4 (t2 )  P57  P5 (t2 )  P67  P6 (t2 )  (2)  P37  P13  P01  P47  P14  P01  P57  ( P15  P01  P25  P02 );  P (t )  P  P (t )  P  P  P  P   8 4 78 7 3 78 37 13 01  P78  P47  P14  P01  P78  P57  P15  P01  P78  P57  P25  P02 . Let's write down the normalizing conditions:  P1 (t1 )  P2 (t1 )  1, т.е. P01  P02  1;  P (t )  P (t )  P (t )  P (t )  1, т.е, P P  P P  P P  P P  1;  3 2 4 2 5 2 6 2 13 01 14 01 15 01 24 02  P7 (t3 )  1, т.е. P37  P13  P01  P47  P14  P01   P  P  P  P  P  P  P  P  1; (3)  57 15 01 57 25 02 25 02  P8 (t4 )  1, т.е. P78  P37  P13  P01  P78  P47  P14  P01   P78  P57  P15  P01  P78  P57  P25  P02  1. The transition probabilities can be estimated based on the experimental data. Then, based on conditions (1), systems (2), and (3), the probabilities of the system states can be found. For the final state, we S 8 specify the minimum threshold P8 , its probability is P8 ( P8  P8 ). If it follows from (1), (2), and (3) that P8  P8 , this means disorganization of functioning of the system under consideration, and it follows that certain conditions shall be created to increase the transition probabilities. In this case, the probabilities of processes in the system shall be considered. 4. Continuous Markovian Chain For a continuous Markovian chain, the time interval from the system in the state S 0 to the state S 8 is considered continuous. Instead, the Pij flow densities ij are considered. The flows can be directed from Si to S j (at j > i) and from S j to Si (at j > i). The density of the reverse flow is specified as  ij . E.g., control commands of higher levels of the system can act as reverse flows. For the Markovian process, we can write a system of Kolmogorov equations. For a sufficiently large interval of observation time t, all the flow densities ij ,  ij and probabilities Pi (i  0,8) can be considered constant values, and all probabilities Pi (t ) (i  0,8) equal to zero. This mode is the system's stationary mode. In this case, we have a homogeneous system of algebraic equations. Thus, the functioning of the lower block associated with the state S 0 in the stationary mode is described by the algebraic equation 163  P0 (01  02 )  P110  P2 20  0. The functioning of the block «Analysis and Correction of the educational and methodological complex for the Discipline» is described by the equation P0 01  P113  P114  P115  P110  P3 31  P4 41  P5 51  0. The other blocks are described in the same way. We Pi (i  0,8) find the sought probabilities by solving a system of algebraic equations for given values ij and  ji (i, j  0,8) . The value is P8 compared with the threshold value P8 . When doing the inequality, it is P8  P necessary to change the densities ij and  ji of transferring flows. 5. Network Representation of the Educational Process Acknowledgements The system shown in Fig. 1 can be represented as a graph (Figure 2), the points of which are the states Si (i  0,8) , and the arcs are the connections between the structural elements of the system indicated in the figure by arrows. Such a graph that has a single initial and a single final point is called a network. The points of the network are called "events", the path passing through these points is called "work". Figure 2: Educational process curve Each state of the Si zero, first and second levels, respectively, i  0,8 also represents a system that is shown as a network. Either conditional transition probabilities or densities of transferring flows can be considered as the "loads" in the network. Any network can be described using certain parameters related to the time characteristics of the process: 1. The early (late) dates of the occurrence of each event; are connected to the transition to this point of the graph (this state) after the completion of all previous works and events. 2. Early (late) dates of the start and end of each work. 3. Time reserves of events and works. 4. Full path (the path connecting the initial point of the network to the final one). 5. The critical path is the longest of the complete paths. The following probabilistic characteristics of the network can be considered: 1. The length of the event path Si is the sum of all the transition probabilities Pkl (or densities kl ) that specify the "loads" on the arcs that determine the path from the initial point to the point Si . 2. The minimum (maximum) probability of the occurrence of the i-th event (reaching the point Si ) is equal to the sum of the transition probabilities related to all the paths leading from the initial point to the point Si . 164 3. Directive probability is the probability of reaching the final state; it is equal to the minimum probability of reaching the final point. 4. A full path is any path connecting the start and endpoints of the network. 5. The critical path on the graph is the path to which the highest total probability is attributed. 6. Reserve probability of occurrence of an event is the maximum probability by which the probability of occurrence of such event can be reduced without reducing the directive probability. The proper functioning of an educational institution should be stable. Stability is understood as the preservation of the main characteristics of the system under external impacts. For an educational system, sustainability is determined by its ability to function under changing conditions of the internal and external environment. Factors of the external environment are, for example, the state of the economy, the development of technology, the socio-cultural sphere, the legal system, the demographic situation, the standard of living of the region. The internal environment includes finance, personnel, organizational culture, image, number of students and quality indicators of students, specialties, the level of qualification of professors and teachers and scientific potential, the organization of a system for monitoring the quality of training, the state of educational laboratory, instrumental, library and sports bases. The system is structurally stable when in case of sufficiently small structural changes, its functioning does not change significantly. Suppose the process is described by a system of equations of the type dxi  ai1 x1  ai 2 x2  ...  ain xn , i  1, n , (4) dt   where all xi - time-varying functions t, aij i, j  1, n - general characteristics of the functioning of the element Sij . In this case, the method developed in [2] can be used to study its stability. Using the matrix A  {aij } , we make up the determinant of the following form: (a11   ) a12 ... a1n a21 (a22   ) ... a2 n  0, (5) an1 an 2 ... (a nn  ) which is set to zero, and an algebraic equation of the n-th order is obtained:  n  b1 n1  ...  bn1  bn  0 . (6) A square Hurwitz square matrix is formed based on the coefficients of this equation:  b1 1 0 0 ... 0   b b b 1 ... 0  Г  3 2 1 ,  b5 b4 b3 b2 ... 0   0 0 0 0 ... b   n where bm  0 at m  n. For the asymptotic stability of the system (4), all the principal diagonal minors of the corresponding Hurwitz matrix are: b1 1 0 b 1 1  b1 ,  2  1 , 3  b3 b2 1 ,...,  n  bn   n 1 , b3 b2 b5 b4 b3 are positive, i.e. i  0 when i  1,2,...,n . Thus, it is possible to determine the stability of the management system of an educational organization. 6. Conclusion 165 In this study, a detailed structural scheme of the educational process has been developed. To describe the processes occurring in the system, the mathematical tool techniques of discrete and continuous Markov chains are used. Formula expressions for calculating the probabilities of the system states are obtained. It is shown how the scheme of the educational process can be represented in the form of a network graph. The study determines the parameters associated with the time characteristics of the process. A new approach to the analysis of the stability of the educational system under consideration has been proposed. The novelty of the proposed work is in the fact that a comprehensive analysis of the educational system (structure, functioning, stability) has been carried out using the method of system analysis. 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