This study proposes a new approach that recovers the system from deadlock states to its former live states, and reaches the same number of states as the original uncontrolled model by adding monitors (and control arcs) with no new problematic siphons. We further propose a lossless approach by coloring some arcs to avoid the material loss.
It has been a hot research topic to compare the effectiveness of new control policies by testing against a wellknown S 3 PR model. So far, only the control policy by Piroddi et al. may reach the optimal number of states among all approaches for a well-known benchmark using a siphon-based approach. The resulting model is a generalized Petri net since some control arcs are weighted, which complicates the system. The only improvement that can be made is to reduce the number of control arcs (by 3), and the number of weighted control arcs (by 9) as we report in this paper. Also the token count is reduced. This is achieved by replacing two monitors with weighted arcs by two new monitors without weighted arcs. INA (Integrated Net Analyzer) analysis indicates that the resulting controlled model is live and reaches the same 21581 states by Piroddi et al. We develop a formal theory for explaining the cause of state losses and providing the foundation for the above improvement model.
Unlike other techniques, Li and Zhou add control nodes and arcs for only elementary siphons greatly reducing the number of control nodes and arcs (implemented by costly hardware of I/O devices and memory) required for deadlock control in Petri net supervisors. Li and Zhou propose that the number of elementary siphons is linear to the size of the net. An elementary siphon can be synthesized from a resource circuit consisting of a set of connected segments. We show that the total number of elementary siphons, |O E |, is upper bounded by the total number of resource places |P R | lower than that min(|P|, |T|) by Li and Zhou where |P| (|T|) is the number of places (transitions) in the net. Also, we claim that the number of elementary siphons |O E | equals that of independent segments (simple paths) in the resource subnet of an S 3 PR (systems of simple sequential processes with resources). Resource circuits for the elementary siphons can be traced out based on a graph-traversal algorithm. During the traversal process, we can also identify independent segments (i.e. their characteristic T-vectors are independent) along with those segments for elementary siphons. This offers us an alternative and yet deeper understanding of the computation of elementary siphons. Also, it allows us to adapt the algorithm to compute elementary siphons in [2] for a subclass of S 3 PR (called S 4 PR) to more complicated S 3 PR that contains weakly dependent siphons.
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 [Formula: see text], 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 [Formula: see text] 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.
Li and Zhou propose to add monitors <i>V<sub>s</sub></i> to elementary siphons <i>S</i> only while controlling the rest of dependent siphons - important for large systems but far from being maximally permissive. The control policy for weakly dependent siphons (WDS) is rather conservative due to some negative terms in the controllability. We show that this is no longer true as can be shown that it has the same controllability as that for strongly dependent siphons
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.