A novel liveness condition for S³PGR²

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 [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.

Policy to cope with deadlocks and livelocks for flexible manufacturing systems using the max′‐controlled new smart siphons

“…The reason is that the output arcs of GCPs are linked to the source transitions of an original plant net model, which may restrict the behaviour permissiveness of the controlled system to some degree, implying that it is need to improve the addition of GCPs in the next research. In addition, this RMIP approach that can solve NSSs associated with deadlocks and livelocks is only a sufficient condition for testing the liveness of S 4 R. As a result, as stated in [26], how to reconstruct the objective function and the corresponding new constraints for the novel MIP method that is a sufficient and necessary condition for the liveness of S 4 R, and like [4,6,7], how to achieve the final live controlled system (N * , M * ) with R = 100% by improving the addition of GCPs, and the extension of this improved DCP to more general classes of Petri nets are the focus of our future research. Table 8 Results of Policy 1 for a nonlive marked S 4 R shown in Fig.…”

“…In a deadlock state, the global or local FMS is blocked or crippled, causing unnecessary cost or catastrophic results. When an FMS is at a livelock state, some processes continuously change their state while the other processes are deadlocked [23][24][25][26]. That is to say, a livelocked system still does some (although not useful) work in which some execution that never make effective progress cycle infinitely.…”

“…Considering the shortcomings of the above SC policies and motivated by conclusion in [26], this paper proposes a novel iterative SC policy that can deal with deadlock and livelock problems. By means of the structural analysis of Petri nets and the concept of smart siphons [31], new constrains [(8) and (9) in Section 3] are generated and then added to the MIP method in [31], called revised MIP (RMIP).…”

“…In general, ZL‐policy is used to verify whether an original net is self‐live or to make it be self‐live, but its computational load is large. Recently, Chao et al [26] indicate that the max′‐controlled condition for problematic siphons is only sufficient for liveness a well‐marked S 4 R. The sufficient and necessary condition for liveness of a well‐marked S 4 R needs to detect a pivot marking and to verify the fact a nonmax′‐marked siphon does not exist under the corresponding marking. It is easy to see that the corresponding computational load is not low.…”

“…It is easy to see that the corresponding computational load is not low. On the basis of the computational efficiency, how to build the novel MIP test that is both sufficient and necessary for liveness of S 4 R remains an open question [26].…”