Semilattice characterization of nonblocking networks

A connecting network is called strictly nonblocking if no call is blocked in any state; it is nonblocking in the wide sense if there exists a rule for routing calls through the network so as to avoid all states in which calls are blocked, and yet still satisfy all demands for connection as they arise, without disturbing calls already present. Characterizations of both senses of nonblocking have been given in previous work, using simple metric and closure topologies defined on the set of states. We give new characterizations based on the natural map γ (·) that carries each state into the assignment it satisfies. This map is a semilattice homomorphism, such that γ (x) ∩ γ(y) ≧ γ (x ∩ y). It turns out that the case of equality in this inequality is very relevant to nonblocking performance. In particular, let a subset X of states be said to have the intersection property if for every x in X and every assignment a there exists y in X such that y realizes a (i.e., γ (y) = a) and γ(x ∩ y) = γ (x) ∩ γ (y). Then a network is nonblocking in the wide sense if and only if some subset of its states has the intersection property, and it is strictly nonblocking if and only if the entire set of states has the intersection property.