The increase in the performance of computer systems (CS) is associated with both scalability and the development of the architecture of the computing elements of the system. Cluster CS, which are scalable, make up 93% of the Top500 supercomputers and are high-performance. At the same time, there is still the problem of efficient and complete use of all available computer resources of the supercomputer and CS for solving user tasks. Failures of elementary machines (nodes, computing modules) reduce the technical and economic efficiency of CS and the efficiency of solving user tasks. Therefore, when planning the process of solving problems, reducing the loss of time to restore CS from failures is an important problem. To quantify the potential capabilities of computer systems, indices of the realizability of solving tasks are used. These indices characterize the quality of the systems, taking into account reliability, time characteristics and service parameters of incoming tasks. The paper proposes a mathematical model of the functioning of a computer system with a buffer memory for group maintenance of a task flow. The mathematical model uses queuing theory methods based on probability theory and systems of differential equations. It should be noted that the method of composing systems of differential equations is simpleenough if the corresponding graph scheme is presented. However, the exact solution of systems of equations and, as a rule, in elementary functions, does not exist, or formulas are difficult to see. Here the solution is obtained in the stationary mode of operation of the queuing system. The indices allowing to estimate the fullness of the buffer memory are calculated. The obtained analytical solutions are simple, can be used for express analysis of the functioning of computer systems.
