Weighted Characteristic P-vector and Deadlock Control of WS3PR

Current deadlock control approaches for S^3PGR^2 (systems of simple sequential processes with general resources requirement) suffer from incorrect or restricted liveness characterization based on the concept of deadly marked siphons (DMS) and max-controlled siphons. Dead transitions may exist when there are no DMS and the net model is in livelock states. A new liveness condition is developed based on the so-called max^*-controlled siphons to replace that of the restrictive max-controlled siphons. A deadlock control policy is further proposed for WS^3PR (weighted S^3PR (systems of simple sequential processes with resources)) by adding control nodes and arcs for elementary siphons only, reducing significantly the number of monitors compared with existing methods. A counter example is shown to indicate that Li's characteristic P-vector must be weighted. The controlled model for WS^3PR is proposed and its liveness property is proved.