SUMMARY
The object of research in this work is the information system that is restored after the failures, consisting of a finite number n of the hardware and software client systems that are connected via the corresponding interface to the server S. This system is a «client-server» model for a modern public notification system in emergency situations. The system provides for the use of services of interactive terrain maps for processing or supplementing data transmitted from mobile devices of the victims, which have already been received by the server and worked through the algorithms programmed in it. In the course of research, mathematical models of the considered information system were constructed and used. The «client-server» information system that is restored after a failure with a finite number of states is approximated. A state graph is obtained that describes its behavior, and a system of Kolmogorov differential equations is also compiled. The direct Laplace transform from a system of differential equations is used. A mathematical model is built to describe the behavior of the system, taking into account the ultimate reliability of the Google Maps service system. Also, to determine the reliability of the model, a special case of the system operation is calculated without taking into account the failure of the client. This is due to the fact that the client part of each potential victim has its own individual characteristics and depends on the type of used device. The behavior of the information system in the presence of failures is analyzed, which is described by a system of differential equations with variable coefficients. This analysis shows that the information system has a fairly high reliability of the main characteristics. Through the use of the proposed mathematical model, it is possible to obtain various characteristics of the reliability of information systems such as «client-server». The solution of the obtained differential equations allows to study these characteristics in a wide range of changes in failure rates and restoration of system components.