Deadlock and liveness characterization for a class of generalized Petri nets

Petri nets (PNs) are widely adopted for modeling flexible manufacturing systems (FMSs) since they are an effective tool for analyzing the latter's dynamic behavior and synthesizing a supervisory controller to make a system deadlock-free. As an important subclass of PNs, the weighted system of simple sequential processes with resources (WS³PR) can be used to model many FMSs well. Based on the initial marking and structural properties of a WS³PR without using marking enumeration and state equations, this paper presents a method to check its liveness. We first define two classes of transitions for a resource subnet of WS³PR, based on which, the relationship between markings and strongly connected resource subnets (SCRSs) is analyzed. Next, functions used to check the liveness of a given WS³PR are developed, which take the full advantage of SCRS trees and WS³PR structure. It is shown that, by the proposed method, the computational complexity for a subclass of WS³PR named k-sharing bounded WS³PR is polynomial. Sufficient conditions to check liveness of a WS³PR are finally established. Two examples are used to illustrate the results.