The master's thesis addresses max-plus algebra as a mathematical framework for modeling and stability analysis of periodic train timetables. Max-plus algebra, which replaces the classical operations of addition and multiplication with the operations of maximum and addition, proves to be particularly suitable for describing systems with a sequential progression of events, where the dynamics are determined by the longest influence in the network. The thesis presents the fundamental algebraic properties of the max-plus space, spectral theory, cyclicity, and the asymptotic behavior of linear max-plus systems, which enables the analysis of stability and long-term dynamics of processes. The theoretical foundations are applied to the development of a railway network model, where a periodic timetable can be naturally described using max-plus algebra. Consequently, this allows for the analysis of the effects of delays, stability margins, traffic density, and the influence of infrastructure capacity constraints. Random max systems are also studied, yielding interesting results.
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