Computation of the number of legal states for petri net-based deadlock prevention problems

Abstract

Petri net (PN) based prevention and control methods are widely studied in the literature to solve deadlock problems in flexible manufacturing systems (FMS). In PN models of FMS suffering from deadlocks, the reachability graph (RG) of the PN model can provide all reachable system states from the initial state, including all legal states, bad states, and deadlock states. A maximally permissive deadlock controller allows the system to reach all legal states exist within the live zone (LZ) that determines the optimal live behavior, while prohibits reaching bad and deadlock states exist within the dead zone. It is necessary to know the exact number of legal states that must be provided by a deadlock controller to determine the behavioral permissiveness of a control policy. Therefore, the number of legal states has been considered as a quality measure for deadlock prevention methods available in the literature. Unfortunately, to date for a given RG of a PN model of an FMS suffering from deadlocks, no study has been reported to provide the number of reachable legal states exist within the LZ of the given RG. In this paper, a method is proposed for the computation of the number of legal states that must be provided by an optimal deadlock prevention policy. The proposed method makes use of the reachability analysis of a given PN model of a deadlock-prone FMS together with the first strongly connected component (SCC) by using INA (Integrated Net Analyzer). The number of legal states computed from the first SCC that includes the initial marking represents the LZ of a RG. The proposed algorithm is implemented as an executable program. The number of legal states of a deadlock controller can easily and correctly be computed by using the proposed method and tool. Several well-known examples of FMS are considered to illustrate the applicability and the effectiveness of the proposed method.