  We consider the scheduling control problem for a family of unitary networks
under heavy traffic, with general interarrival and service times, probabilistic
routing and infinite horizon discounted linear holding cost. A natural
nonanticipativity condition for admissibility of control policies is
introduced. The condition is seen to hold for a broad class of problems. Using
this formulation of admissible controls and a time-transformation technique, we
establish that the infimum of the cost for the network control problem over all
admissible sequencing control policies is asymptotically bounded below by the
value function of an associated diffusion control problem (the Brownian control
problem). This result provides a useful bound on the best achievable
performance for any admissible control policy for a wide class of networks.
