A novel liveness condition for S 3 PGR 2

Abstract

Systems of simple sequential processes with resources (S³PR), modeled by ordinary Petri nets (OPNs) are live if and only if no siphons (a set of places) can ever become empty of tokens. Systems of simple sequential processes with general resources requirement (S³PGR²), modeled by general Petri nets (GPNs), are a generalization of S³PR. It has been a hot research topic to find the sufficient and necessary condition of liveness for S³PGR². When an OPN (respectively, GPN) is deadlocked, the set of all unmarked (respectively, non-max-marked) places forms a siphon, which is said to be deadly marked. However, when an S³PGR² is livelocked but not deadlocked, e.g., both live and dead transitions exist, the set of non-max-marked places, denoted by $\vartheta$ , may not form a siphon. Thus, it is limited using deadly marked or max-controlled siphons to characterize liveness of S³PGR². The first author proposed earlier max’-controlled siphons to characterize liveness of S³PGR². However, the presence of a set of non-max’-marked places is not a sufficient condition for an S³PGR² to be non-live and cannot be relied on to devise mixed integer programming (MIP) testing. In this paper, we present the sufficient and necessary liveness condition of S³PGR². We define pivot markings, under which we prove that $\vartheta$ forms a siphon. To detect livelocks, the number of disabled arcs must be maximal under the pivot marking. This result is useful to reorient the MIP test of S³PR for S³PGR².