Modeling and Control of Weight-Balanced Timed Event Graphs in Dioids

The class of Timed Event Graphs (TEGs) has widely been studied for the last 30 years thanks to an algebraic approach known as the theory of Max-Plus linear systems. In particular, the modeling of TEGs via formal power series has led to input-output descriptions for which some model matching control problems have been solved. In the context of manufacturing applications, the controllers obtained by these approaches have the effect of regulating material flows in order to decrease internal congestions and intermediate stocks. The objective of this work is to extend the class of systems for which a similar control synthesis is possible. To this end, we define first a subclass of timed Petri nets that we call Balanced Timed and Weighted Event Graphs (B-TWEGs). B-TWEGs can model synchronisation and delays (B-TWEGs contains TEGs) and can also describe some dynamic phenomena such as batching and event duplications. Their behavior is described by some rational compositions of four elementary operators γ n , δ t , μm and βb on a dioid of formal power series. Then, we show that the series associated to B-TWEGs have a three dimensional graphical representation with a property of ultimate periodicity. This modeling allows us to show that B-TWEGs can be handled thanks to finite and canonical forms. Therefore, the existing results on control synthesis, in particular the model matching control problem, have a natural application in that framework

Performance evaluation of an emergency call center: tropical polynomial systems applied to timed Petri nets

We analyze a timed Petri net model of an emergency call center which processes calls with different levels of priority. The counter variables of the Petri net represent the cumulated number of events as a function of time. We show that these variables are determined by a piecewise linear dynamical system. We also prove that computing the stationary regimes of the associated fluid dynamics reduces to solving a polynomial system over a tropical (min-plus) semifield of germs. This leads to explicit formul{\ae} expressing the throughput of the fluid system as a piecewise linear function of the resources, revealing the existence of different congestion phases. Numerical experiments show that the analysis of the fluid dynamics yields a good approximation of the real throughput.Comment: 21 pages, 4 figures. A shorter version can be found in the proceedings of the conference FORMATS 201