Abstract
Liveness is one of the most important properties in Petri net, which is a powerful tool for modeling of discrete event systems.
As an extended class of Petri nets, timed Petri nets with finite firing duration of transitions have a wide range of applications, such as scheduling problems in FMS, parallel computing systems and so on. The earliest firing rule for timed Petri net model offers an easy and effective way for solving sub-optimization problems. On the other hand, it is often the case in practical problems that some tasks share processors with other tasks. Such processors can be modeled by ‘resource places’ in Petri net.
In this paper, we treat liveness problems of the timed Petri nets which have shared resource places and obey the earliest firing rule. In general, liveness of a usual Petri net is neither necessary nor sufficient condition for liveness of the Petri net with shared resource places under the earliest firing rule. The relations between above two cases are clarified for some subclasses.
First, considering timed Petri nets with uniform firing durations, we show that if the net has a POC (partially ordered condition) structure and if the net is live, the net with arbitrary resource places is live under the earliest firing rule. The converse is not always true, but if the underlying net is SMA net, the two notions of liveness are equivalent.
For timed Petri nets with arbitrary firing durations, providing some additional conditions on resource places and the underlying net, we show that liveness is preserved under the earliest firing rule (1) if addition of resource places preserves POC structure, or (2) if resource places are connected to “fair” transitions. Some typical models in scheduling problems satisfy these conditions.
