Modeling and Analysis of the Metro-bus Public Transport System in Mexico City using Timed Event Petri Nets and Max-Plus Algebra

Abstract: In this paper an algorithm for computing a generalized eigenmode of reducible regular ma-trices over the max-plus algebra is applied to the Metro-bus public transport system in Mexico city. A timed event Petri net model is constructed from the data table that characterizes the transport system. A max-plus recurrence equation, with a reducible and regular matrix, is associated to the transport system timed event Petri net. Next, given the reducible and regular matrix of finite size i.e., a matrix such that in each one of its rows has at least one finite element and whose communication graph is not strongly connected, the problem consists in giving an algorithm which will tell us how to compute its generalized eigenmode over the max plus algebra, which indeed has an idempotent semiring, or also called dioid, mathematical structure. The notion of generalized eigenmode, as its name says, results to be a genaralization of the notion of eigenvalues and eigenvectors for the case when the matrix under study is irreducible i.e., has a communication graph which is strongly connected. The solution to the problem is achieved by studying some type of recurrence equations. In fact, by transforming the re-ducible regular matrix into a block upper triangular form, called normal form, and considering a very specific recurrence equation, an explicit mathematical characterization is obtained, upon which the al-gorithm is constructed. The generalized eigenmode obtained sets a timetable for the transport system