Modified definition of characteristic P-vector for elementary siphons in a class of Petri nets

Li and Zhou developed a method to compute elementary siphons and indicated that it may be extended to weighted resource allocation systems. The author shows a better way to find elementary siphons for weighted (SPR)-P-3 (systems of simple sequential processes with resources) or (WSPR)-P-3 via an example where there is no dependent siphon based on the approach by Li and Zhou. Yet, the author is able to locate one; thus, reducing the number of monitors required. Redefining the characteristic P-vector by weighting each component by that of a P-invariant, the author shows via a theorem and an example that an (SPR)-P-3 and its weighted (SPR)-P-3 have the same sets of elementary, dependent siphons and systems of equations of characteristic T-vectors. Applying their siphon-synthesis theory, the time to find elementary siphons for (WSPR)-P-3 is reduced from exponential to polynomial. The author also derives the controllability condition for a (WSPR)-P-3 and discuss how to extend to S(3)PGR(2) (systems of simple sequential processes with general resources requirement).