Priced Timed Petri Nets

TL;DR

This paper introduces priced timed Petri nets, a model combining timing and cost constraints, and analyzes the computability of minimal costs for reaching control states under various cost conditions.

Contribution

It extends existing models by incorporating real-valued clocks and costs, providing algorithms for computing minimal costs with non-negative costs and proving undecidability with negative costs.

Findings

Infimum of costs is computable with non-negative costs.

Reachability with zero cost is undecidable with negative costs.

Model generalizes priced timed automata and unbounded priced Petri nets.

Abstract

We consider priced timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Furthermore, our cost model assigns token storage costs per time unit to places, and firing costs to transitions. This general model strictly subsumes both priced timed automata and unbounded priced Petri nets. We study the cost of computations that reach a given control-state. In general, a computation with minimal cost may not exist, due to strict inequalities in the time constraints. However, we show that the infimum of the costs to reach a given control-state is computable in the case where all place and transition costs are non-negative. On the other hand, if negative costs are allowed, then the question whether a given control-state is reachable with zero overall cost becomes undecidable. In fact, this negative result holds even in the simpler case of discrete time (i.e., integer-valued clocks).