Analysis of Mathematical Models of Functioning Scalar Multiprocessor Systems

: pp. 66 - 78
Lviv Polytechnic National University
National Aviation University
National Aviation University

The article is devoted to the research of mathematical models of functioning of complex systems. The correspondence of the results of the study with the qualitative idea about the behavior of the system, saturation of the system with increasing number of users. 

A model of a homogeneous complex system with priority (flow) of large tasks is constructed. The studies of the functioning of a homogeneous complex system in processing the flows of complex tasks have been carried out on the assumption that a Poisson flow of programs (P-programs) with a certain frequency enters the system of N elementary personal electronic computers with additional infinite external memory. New approach to the study of the functioning of complex systems is proposed. The system is assumed to have different types of resources and applications that require different combinations of these resources. Thus, the paper proposes a rather detailed model of the system, which is called a multi-resource system of massive maintenance. A technique for determining the power limit of a multi-resource mass service system is proposed. This technique allows you to determine the power limits for a system with fixed input stream characteristics. The model is proposed that allows to establish the minimum required number of sources of queries at a given number of processing zones of the work field. When resources are redistributed tactfully, the model described can be used to select multiprogramming for both multiprocessor (processor allocation between tasks) and multiprogram single-processor computing systems (allocation of RAM between tasks).

The method of calculating the number of variants of assignments of tasks in a computer system within the given structure or set of structures, as well as the method of determining the probable characteristics of homogeneous complex systems in the mode of memorizing the processing of complex problems in different disciplines of distribution are offered. It is suitable for the design characteristics of modern homogeneous complex systems, which should contain hundreds and thousands of mini-processors and serve hundreds of users.

The analysis of mathematical models concerning the functioning of complex systems is carried out, which allows the choice of a particular system to solve a specific problem. The results obtained are consistent with the perceptions of system behavior and system saturation as the number of users increases. In addition, the probability of solving all problems by the system is revealed, which allows to predict the performance of homogeneous complex systems and to be used in choosing the parameters of the designed systems.

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