Controllability for siphons in S3PGR2

Insufficiently marked siphons cause deadlocks in FMS. To avoid deadlocks, monitors and control arcs are added upon these siphons. The number of such siphons grows exponentially with the size of the net modeling the FMS. Li et al. propose to add monitors only to elementary siphons (linear complexity) and indicated that it may be extended to weighted resource allocation systems (RAS). Computational efforts are required to select elementary siphons among all problematic siphons and to express the characteristic T-vector of each dependent siphon in terms of the linear summation of those of elementary siphons. We discovered earlier that elementary (resp. dependent, called compound), called basic siphons in an S³PR (systems of simple sequential processes with resources) might be synthesized from elementary (resp. compound) resource circuits. This has the advantage of avoiding the above computation. However, this no longer holds for those in an S³PGR². This paper revises the definition of elementary siphons so that basic and compound siphons in an S³PGR² (systems of simple sequential processes with general resource requirements) remain to be elementary and dependent siphons, respectively. In addition, we will derive the exact controllability (both sufficient and necessary) so that the subsequent IP (Integer programming) test can be eliminated.