Elementary siphons of Petri nets and their application to deadlock prevention in flexible manufacturing systems

TLDR

Deadlock prevention in flexible manufacturing systems traditionally relies on adding control places to all minimal siphons, but the exponential growth of minimal siphons with net size leads to excessive added places and model complexity, while elementary siphons form a proper subset of minimal siphons. This study aims to reduce the number of added control places needed for deadlock prevention without sacrificing effectiveness. The authors introduce elementary siphons and propose adding a single control place to each to keep it marked, demonstrating the approach on a flexible manufacturing system example. The method achieves deadlock prevention with far fewer control places than existing techniques, as shown by its smaller cardinality in large Petri nets and its superior performance in the illustrative example.

Abstract

A variety of important Petri net-based methods to prevent deadlocks arising in flexible manufacturing systems (FMS) are to add some control places and related arcs to strict minimal siphons (SMS) such that no siphon can be emptied. Since the number of minimal siphons grows in general exponentially with respect to a Petri net size, their disadvantages lie in that they often add too many additional places to the net, thereby making the resulting net model much more complex than the original one. This paper explores ways to minimize the new additions of places while achieving the same control purpose. It proposes for the first time the concept of elementary siphons that are a special class of siphons. The set of elementary siphons in a Petri net is generally a proper subset of the set of all SMS. Its smaller cardinality becomes evident in large Petri net models. This paper proves that by adding a control place for each elementary siphon to make sure that it is marked, deadlock can be successfully prevented. Compared with the existing methods, the new method requires a much smaller number of control places and, therefore, is suitable for large-scale Petri nets. An FMS example is used to illustrate the proposed concepts and policy, and show the significant advantage over the previous methods.