Timed Hierarchical Object-Oriented Petri Net

Petri nets (Murata, 1989) (Peterson, 1991) have been widely used to model various discrete event systems (Moody & Antsaklis, 1998). Characterized as concurrent, asynchronous, distributed, parallel, nondeterministic, and/or stochastic (Murata, 1989), Petri nets have gained more and more applications. However, when they are used to analyze and model systems of different domains, the shortages of this kind of formal method still exist. Basic Petri nets lack temporal knowledge description, so they have failed to describe the temporal constraints in time critical or time dependent systems. The introduction of temporal knowledge into Petri nets has increased not only the modeling power but also the model complexity (Wang et al. 2000). The improved models of Petri nets (Wang, 1998) include Timed Petri Net (Ramchandi, 1974), Stochastic Timed Petri Net (Florin etc al., 1991) and Time Petri Net (TPNs) (Merlin & Farber, 1976). In TPNs (Merlin & Farber, 1976), each bar has two times specified. The first time denotes the minimal time that must elapse from the time that all the input conditions of a bar are enabled until this bar can fire. The other time denotes the maximum time that the input conditions can be enabled and the bar does not fire. After this time, the bar must fire. In general, these two times give some measures of minimal and maximal execution times of the bars. The reachability (coverability) analysis is one of the main analysis methods for Petri nets (Murata, 1989), in which the coverability tree is always used. It permits the automatic translation of behavioral specification models into a state transition graph made up of a set of states, a set of actions, and a succession relation associating states through actions (Bucci & Vivario, 1995). That is to say, it involves essentially the enumeration of all reachable markings or their coverable markings. This representation makes such properties as deadlock and reachability (Zhou, 1995) explicit, and allows the automatic verification of ordering relationships among task execution times (Tsai et al., 1995). Although the reachability analysis method can be used for all nets, it is only limited to “small” nets due to the complexity of the state-space explosion. The same thing also happens in the analysis of TPN models. Sloan et al. (Sloan & Buy, 1996) developed several reduction rules for TPN analysis that work at an individual transition level. These reduction rules help to reduce the complexity of TPN analysis to some extent. However, it is not a trivial work to automatically search the preconditions of applying these reduction rules for a