Enumeration of lost states of a suboptimal control model of a well-known S³PR

Deadlocks halt a system completely causing a significant financial loss to a company. To resolve this problem, deadlock prevention (by adding monitors to problematic siphons) has been quite a popular research. Uzam and Zhou applied region analysis to a well-known S³PR to achieve a near-maximum permissive control policy. However, they do not list the lost states, which is essential to improve the control model. The lost states can be obtained by reachability analysis, which is a rather tedious process. Without theory, one could waste much time failing to reach more states and there is no effective solution so far in the literature. Thus, it is important to find out the condition where more states can be reached. If no more states can be reached, one should simply stop and remain satisfied with the suboptimal model obtained or employ weighted control arcs to reach more states. It is desirable to compute the gain of states without the costly reachability analysis when an alternative control policy is employed. It is interesting to explore which live states are lost in the first-met bad marking (FBM) method, which has not yet been available in the literature. This study presents the very first method to compute all lost states based on invariant without reachability analysis.