Weighted Characteristic P-vector and Deadlock Control of WS3PR

Current deadlock control approaches for S(3)PGR(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 (WSPR)-P-3 (weighted (SPR)-P-3 (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 (WSPR)-P-3 is proposed and its liveness property is proved.