Construct the closed-form solution of A-net of Petri nets by case study

Abstract

After our researches on the effect of a non-sharing resource in a kth order which is the concept of customization manufacturing, in this article we extend the research on the closed-form solution of control-related states to the so-called Anet which has one top non-sharing circle subnet connected to the idle place of left process in a deficient kth order system and is the fundamental model of different productions sharing the same common parts in manufacturing. The formulas just are depended on the parameter k and states' function of top non-sharing circle subnet for a subclass of nets with k sharing resources. By combining the concept of the partial deadlock avoidance/prevention policy, the moment to launch resource (controller) allocation based on the current state, and the construction of closed-form solution for deficient kth order system, it can realize the concept of dynamic non-sharing processes' allocation.

Topological reverse mirroring: a new efficient knowledge-based methodology of reachability analysis for Petri nets

“…Figure 3(a,b)). According to the concept of a Complete reachability graph, Chao (2014) split the reachability graph of a PN N into reachable (from the initial state), live (reachable to the initial state), forbidden (reachable to the livelock (Chao & Yu, 2017) or deadlock states only), livelock (Chao & Yu, 2017) (reachable to the livelock state only), deadlock (reachable to no state), non-reachable (from the initial state) and non-reachable+ empty-siphon states (both non-reachable states in a N and the rev(N)). Due to the independence between the set of live and forbidden states, we have F = R -L, where R, F and L are the number of reachable, forbidden and live states of a PN.…”

“…The formulas are consistent with the reachability analysis using the INA (Integrated Net Analyzer) tool (Starke, 1992). Chao and Yu (2017) showed that the Deficient k-th order system is the essential element of a k-th order system with a non-sharing sub-net (NNS) (as shown in Figure 5) because the NNS will share some tokens in the idle place of left-hand-side process. Beside, Chao and Yu showed that there are so-called livelock states that presently INA cannot detect in such a system.…”

“…Even mixed integer programming (MIP) method applied for system-control applications in PNs (Li, Wu, et al, 2012;Nazeem et al, 2011;Wang et al, 2020), the non-deterministic polynomial-time hard (NP-hard) characteristic of the MIP (Schrijver, 1998) still limits the size of real-time systems modelled in real application. Chao (2014) broke through this barrier earlier by proposing that we can apply mathematics in terms of closed-form solutions (CFSs) to solve the reachability problem of PNs, and then constructing a solid theoretical framework based on graph theory to derive the CFS of the number of reachable, live, forbidden, deadlock, livelock (added by Chao and Yu (2017)), non-reachable and CONTACT Tsung Hsien Yu yutsunghsien@gmail.com No. 59, Nongquan Rd., Yilan City, Yilan County 260006, Taiwan (R.O.C.…”