Comparing digraph and Petri net approaches to deadlock avoidance in FMS

Flexible manufacturing systems (FMSs) are modern production facilities with easy adaptability to variable production plans and goals. These systems may exhibit deadlock situations occurring when a circular wait arises because each piece in a set requires a resource currently held by another job in the same set. Several authors have proposed different policies to control resource allocation in order to avoid deadlock problems. These approaches are mainly based on some formal models of manufacturing systems, such as Petri nets (PNs), directed graphs, etc. Since they describe various peculiarities of the FMS operation in a modular and systematic way, PNs are the most extensively used tool to model such systems. On the other hand, digraphs are more synthetic than PNs because their vertices are just the system resources. So, digraphs describe the interactions between jobs and resources only, while neglecting other details on the system operation. The aim of this paper is to show the tight connections between the two approaches to the deadlock problem, by proposing a unitary framework that links graph-theoretic and PN models and results. In this context, we establish a direct correspondence between the structural elements of the PN (empty siphons) and those of the digraphs (maximal-weight zero-outdegree strong components) characterizing a deadlock occurrence. The paper also shows that the avoidance policies derived from digraphs can be implemented by controlled PNs.