Comparison and aggregation of max-plus linear systems

We study linear systems in the max-plus algebra, where the basic operations are maximum and addition. We define a preorder to compare the state vectors of max-plus linear systems with the same dimension. We provide two algebraic methods to get bounds (with respect to this preorder) on the state vectors of a lumped max-plus linear system. The first method is based on the strong lumpability. The second method is based on the coherency property, which also allows one to provide bounds on the state vectors of the original linear system from those for the lumped system. We provide the algorithms to compute all the proposed bounds. We show that they can be used for models with a large state index set by means of a time and space complexity analysis.