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Petri nets are an important and popular tool to model and analyze deadlocks in automated manufacturing systems. The state space of a Petri net model can be divided into two disjoint parts: a live-zone and a dead-zone. A first-met bad marking (FBM) is a marking in the dead-zone, representing the very first entry from the live-zone to the dead-zone, and the calculation of FBMs to a large extent contributes to the complexity of designing optimal liveness-enforcing supervisors. Most existing studies have to fully enumerate the reachable markings of a Petri net model to obtain the FBMs, which exacerbates the computational overheads. This paper first explores a variation mechanism of calculating FBMs with respect to the resource capacity in a class of S<sup>3</sup>PR (Systems of Simple Sequential Processes with Resources) from the structural analysis perspective, which contains a <inline-formula> <tex-math notation="LaTeX">$\xi $ </tex-math></inline-formula>-resource. More generally, for the class of S<sup>3</sup>PR with an <inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula>-resource as defined in this paper, the FBMs can be calculated in an algebraic way by a customized structural analysis technique without enumerating all the reachable markings. Finally, the variation mechanism of calculating FBMs is revealed for these considered classes of Petri net models. Examples are given to demonstrate the proposed method.

On Algebraic Identification of Critical States for Deadlock Control in Automated Manufacturing Systems Modeled With Petri Nets