Given recent events, training during quarantine can only take place remotely. To ensure quality training, communication must be seamless. To do this, the network must function smoothly. The solution to this problem is functionally stable networks that allow uninterrupted transmission of information due to redundancy. An important issue is the definition of redundancy. To solve this problem, the article considers the method of synthesis of the structure of the distance learning system. The method of synthesis of network structure used for providing distance learning by the criterion of maximum functional stability on the basis of the introduction of correcting communication lines is offered. With this method, you can develop tools for self-recovery of distributed software, taking into account the characteristics of disparate computer resources through the use of redundancy. This will allow you to develop functionally stable software systems, significantly reduce the recovery time of such systems after or in the event of possible failures. To increase the efficiency of the developed method, the mathematical model of the hyper network based on two hypergraphs was improved, which allows taking into account different requirements for the quality of the network.

The current pace of development of information technology has created the preconditions for the emergence of a wide range of tools for providing educational services using distance learning technologies. This is confirmed by the intensification of the use of distance learning systems in the conditions of sanitary and epidemiological restrictions and the need for acute cost savings.
The existing scientific and methodological apparatus for studying the quality of functioning of distance learning systems is mostly based on approaches to separate assessment of the effectiveness of their elements and the relevant quality indicators. This limits the ability to take into account important factors in the decision-making process and requires comprehensive consideration of the contributions of relevant subsystems to the functioning of the distance learning system.
In order to solve this problem, the article presents a comprehensive methodology for assessing the effectiveness of distance learning system, based on methods of probability theory and hierarchy analysis and describes the patterns of influence on the effectiveness of distance learning system of importance and contributions of its subsystems. Comparative analysis of the data obtained by monitoring and forecasting performance indicators based on the proposed method, shows an increase in the reliability of the assessment within 15-18%, which in contrast to the existing reduction of standard deviation of performance indicators by an average of 26% and ensures the adequacy of the results within certain assumptions and hypotheses.
In such conditions, the choice of an appropriate option for the construction of the DN system is ensured by the reliability of the forecast of the results of its operation in the range of 82-85%, which is high enough to make appropriate decisions.
The value of this study lies in the possibility of using the tested scientific and methodological apparatus in forecasting the outcome of the system and saving material, financial and human resources in the process of implementing the relevant recommendations in practice. This fact makes it possible to eliminate limitations in the practice of building distance learning systems and creates a new opportunity to cover a wider range of factors that affect the quality of operation.The application of this technique makes it possible to predict the results of the joint operation of the relevant subsystems of the distance learning system, taking into account their contribution to the overall result.