Dynamic Behavior of Discrete Events Systems in Dioid Algebra

Unlike natural systems, which obey the laws of physics, discrete-event systems (DES) are generally man-made systems whose behavior cannot be described by continuous functions. They are characterized by discrete dynamics evolving in a finite countable set.

This class of systems includes, for example :

• manufacturing systems, where material flows are studied;

• transport systems ;

• computer systems.

To study these systems, it is necessary to have models capable of taking into account all their dynamic characteristics, which are often complex in nature. However, the phenomena brought into play by SEDs, and responsible for their behavior, are numerous and diverse in nature: sequential or simultaneous tasks, timed or not, synchronized or concurrent. This diversity of phenomena makes it impossible to describe all SEDs with a single model that is both faithful to reality and mathematically exploitable.

Several modelling concepts have been developed: for example, Markov chains for the control of stochastic processes. , or deterministic Petri nets for resource optimization ( ).

Certain sub-classes of SEDs, involving only synchronization and delay phenomena, can be modeled by a class of special Petri nets, called "Timed Event Graphs" (TEGs). It has been shown that the latter admit a linear representation in a particular algebraic structure, called "dioid algebra".