*Result*: ІНТЕГРАЦІЯ МЕТОДІВ ДОСЛІДЖЕННЯ ОПЕРАЦІЙ І ДИСКРЕТНОЇ МАТЕМАТИКИ ДЛЯ ОПТИМІЗАЦІЇ ПРОЦЕСІВ У ПРОГРАМНОМУ ЗАБЕЗПЕЧЕННІ ОПЕРАЦІЙНИХ СИСТЕМ.

*The article presents an approach to optimizing processes in operating system software based on the integration of operations research (OR) and discrete mathematics (DM) methods. It is shown that modern operating systems operate in high-parallelism environments where processes compete for limited resources, which necessitates the search for optimal allocation strategies. Game theory, as a component of OR, provides a framework for modeling strategic interactions among processes, while discrete mathematics offers formal tools for describing structural relationships between them. A game-theoretic model of process scheduling optimization is proposed, where each process is considered a player with a utility function depending on execution time, resource volume, and priority. The application of Nash equilibrium enables finding a stable resource distribution without centralized control. A generalized mathematical model is developed in the form of a directed graph, implementing optimal scheduling algorithms based on dynamic programming, linear optimization, and graph theory. Comparative analysis between classical scheduling algorithms (FCFS, Round Robin, Priority) and the proposed game-optimization model demonstrates reduced average waiting time and improved CPU utilization. The obtained results confirm the efficiency of the integrated approach and highlight its potential application in modeling cloud and distributed systems. The proposed methodology can also be used in educational practice when teaching courses such as Discrete Mathematics, Operations Research, and Operating Systems. [ABSTRACT FROM AUTHOR]

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