Controlled stochastic networks in heavy traffic: Convergence of value functions

Scheduling control problems for a family of unitary networks under heavy trac with general interarrival and service times, probabilistic routing and an innite horizon discounted linear holding cost are studied. Diusion control problems, that have been proposed as approximate models for the study of these critically loaded controlled stochastic networks, can be regarded as formal scaling limits of such stochastic systems. However, to date, a rigorous limit theory that justies the use of such approximations for a general family of controlled networks has been lacking. It is shown that, under broad conditions, the value function of the suitably scaled network control problem converges to that of the associated diusion control problem. This scaling limit result, in addition to giving a precise mathematical basis for the above approximation approach, suggests a general strategy for constructing near optimal controls for the physical stochastic networks by solving the associated diusion control problem.