On Supervisory Policies that Enforce Global Fairness and Bounded Fairness in Partially Controlled Petri Nets

In this paper we consider the notions of global-fairness (G-fairness) and bounded-fairness (B-fairness) for arbitrary Petri nets (PNs). G-fairness in a PN guarantees every transition occurs infinitely often in every valid firing sequence of infinite length. B-fairness guarantees a bound on the number of times a transition in the PN can fire without some transition firing at least once. These properties are guaranteed without recourse to assumptions on firing time distributions or contention resolution policies. We present a necessary and sufficient condition for the existence of supervisory policies that enforce G-fairness and B-fairness along with various observations on the closure properties of policies that enforce these notions of fairness in controlled PNs with a (possibly) non-empty set of uncontrollable transitions. We also derive a necessary and sufficient condition that guarantees a minimally restrictive supervisor that enforces these notions of fairness for bounded PNs. These results are illustrated via examples.