Petri Nets with Time and Cost

Petri nets [13, 12] are a widely used model for the study and analysis of concurrent systems. Many different formalisms have been proposed which extend Petri nets with clocks and real-time constraints, leading to various definitions of Timed Petri nets (TPNs) (see [10, 6] for surveys). In parallel, there have been several works on extending the model of timed automata [4] with prices (weights) (see e.g., [5, 11, 8]). Weighted timed automata are suitable models for embedded systems, where we have to take into consideration the fact that the behavior of the system may be constrained by the consumption of different types of resources. Concretely, weighted timed automata extend classical timed automata with a cost function Cost that maps every location and every transition to a nonnegative integer (or rational) number. For a transition, Cost gives the cost of performing the transition. For a location, Cost gives the cost per time unit for staying in the location. In this manner, we can define, for each computation of the system, the accumulated cost of staying in locations and performing transitions along the computation. In this tutorial, we recall, through a sequence of examples, a very expressive model, introduced in [2], that subsumes the above models. Priced Timed Petri Nets (PTPN) are a generalization of classic Petri nets [13] with real-valued (i.e., continuous-time) clocks, real-time constraints, and prices for computations. In a PTPN, each token is equipped with a real-valued clock, representing the age of the token. The firing conditions of a transition include the usual ones for Petri nets. Additionally, each arc between a place and a transition is labeled with a time-interval whose bounds are natural numbers (or possibly ∞ as upper bound). These intervals can be open, closed or half open. Like in timed automata, this is used to encode strict or non-strict inequalities that describe constraints on the real-valued clocks. When firing a transition, tokens which are removed from or added to places must have ages lying in the intervals of the corresponding transition arcs. We assign a cost to computations via a cost function Cost that maps transitions and places of the Petri net to natural numbers. For a transition t, Cost(t) gives the cost of performing the transition, while for a place p, Cost(p) gives the cost per time unit per token in the place. The total cost of a computation is given by the sum of all costs of fired transitions plus the storage costs for storing certain numbers of tokens in certain places for certain times during the computation. Like in priced timed automata, having integers as costs and time bounds is not a restriction, because the case of rational numbers can be reduced to the integer case.