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.
The main feature of scalable computer systems is modularity. Increasing performance in such systems is achieved by increasing the same type of elements, elementary machines (EM, for example, a computing node). As a result of failures, the system performance is changed. Thus, scalability of computer systems (CS), on the one hand, increases performance, but on the other hand, computer resource growth exacerbates the problem of reliability and increases the complexity of organizing effective functioning. Analysis of reliability and potential capabilities of computing systems is still an urgent problem. For quantitative analysis of the functioning of scalable computing systems, robustness indices related to reliability are used. For example, indices of potential robustness of CS take into account the fact that all operable elementary machines are used in solving tasks, the number of which (EM) changes over time as a result of failures and recoveries. When analyzing reliability, models based on the theory of Markov processes and Queuing theory (QT) are popular in the theory of computing systems. Most QT analytical models do not consider the switching time (reconfiguration) in a separate parameter, due to the complexity of thesolution. Usually, models are simplified by the fact that the recovery time and switch combined in a single parameter. Analytical solutions of a system of differential equations with three parameters (failure, recovery, and switching) for calculating reliability and potential robustness are obtained on the example of the QT model. This allows the user to determine whether the switching time should be taken into account. Also it is shown that solutions of the three-parameter model are reduced to solutions of the two-parameter model if the switching time is not taken into consideration.
