Heavy-Traffic Optimality of a Stochastic Network under Utility-Maximizing Resource Control

We study a stochastic network that consists of a set of servers processing multiple classes of jobs. Each class of jobs requires a concurrent occupancy of several servers while being processed, and each server is shared among the job classes in a head-of-the-line processor-sharing mechanism. The allocation of the service capacities is a real-time control mechanism: in each network state, the control is the solution to an optimization problem that maximizes a general utility function. Whereas this resource control optimizes in a ``greedy'' fashion, with respect to each state, we establish its asymptotic optimality in terms of (a) deriving the fluid and diffusion limits of the network under this control, and (b) identifying a cost function that is minimized in the diffusion limit, along with a characterization of the so-called fixed point state of the network.