=Paper=
{{Paper
|id=Vol-3057/paper17.pdf
|storemode=property
|title=System Analysis of Educational Process
|pdfUrl=https://ceur-ws.org/Vol-3057/paper17.pdf
|volume=Vol-3057
|authors=Antonina V. Ganicheva,Aleksey V. Ganichev
}}
==System Analysis of Educational Process==
https://ceur-ws.org/Vol-3057/paper17.pdf
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 ) P110 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 P113 P114 P115 P110 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 n1 ... bn1 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.
A systematic description of the educational process can be widely used to find control flows and
calculate indicators that determine the stable functioning of not only educational systems but also other
social and economic systems for arranging effective management.
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