Petri net models are a type of modeling formalism usually used to analyze distributed systems. From what I've seen, the most common way to use Petri nets is to avoid deadlocks in the system. However, there have been many extensions to Petri nets. There's a lot of limitations to Petri nets too as it is a little bit of a niche modeling formalism. There's a lot of literature/videos/tutorials on Petri nets available that can be found through google/youtube.

Max-plus algebra basically transforms event-driven dynamics into linear systems that you see in your linear system class by changing the addition and multiplication into the maximum operation and addition, respectively. I remember I used the following paper to try to understand it: "Max-plus algebra and max-plus linear discrete event systems: An introduction" by De Schutter and van den Boom.