On Modeling and Control of Discrete Timed Event Graphs with Multipliers Using (min, +) Algebra

Petri nets are widely used to model and analyze discrete-event systems. We consider in this paper timed event graphs1 with multipliers (TEGM’s). Such graphs are well adapted for modeling synchronization and saturation phenomena. The use of multipliers associated with arcs is natural to model a large number of systems, for example when the achievement of a specific task requires several units of a same resource, or when an assembly operation requires several units of a same part. Note that TEGM’s can not be easily transformed into (ordinary) TEG’s. It turns out that the proposed transformation methods suppose that graphs are strongly connected under particular server semantics hypothesis (single server in (Munier, 1993), or infinite server in (Nakamura and Silva, 1999)) and lead to a duplication of transitions and places. This paper deals with just in time control, i.e., fire input transitions at the latest so that the firings of output transitions occur at the latest before the desired ones. In a production context, such a control input minimizes the work in process while satisfying the customer demand. To our knowledge, works on this tracking problem only concern timed event graphs without multipliers (Baccelli et al., 1992, §5.6), (Cohen et al., 1989), (Cottenceau et al., 2001).