Timed-event graphs with multipliers and homogeneous min-plus systems

The authors study fluid analogues of a subclass of Petri nets, called fluid timed-event graphs with multipliers, which are a timed extension of weighted T-systems studied in the Petri net literature. These event graphs can be studied naturally, with a new algebra, analogous to the min-plus algebra, but defined on piecewise linear concave increasing functions, endowed with the pointwise minimum as addition and the composition of functions as multiplication. A subclass of dynamical systems in this algebra, which have a property of homogeneity, can be reduced to standard min-plus linear systems after a change of counting units. The authors give a necessary and sufficient condition under which a fluid timed-event graph with multipliers can be reduced to a fluid timed-event graph without multipliers. In the fluid case, this class corresponds to the so-called expansible timed-event graphs with multipliers of Munier, or to conservative weighted T-systems. The change of variable is called here a potential. Its restriction to the transitions nodes of the event graph is a T-semiflow.