Heavy-traffic convergence of the prelimit modified control problem

Prove that the optimal objective value V^r(0) of the rth prelimit stochastic processing network control problem with drift-rate bounds of order √r converges to the optimal value V(0) of the modified Brownian control problem as r tends to infinity.

Background

The paper constructs a modified Brownian control problem by replacing the queueing network’s scaled deviation controls with controls consisting of a bounded drift-rate component and a boundary term that acts when a buffer is empty. The authors argue that the drift-rate bounds should scale proportionally to √r in the rth queueing system in order to reflect the original capacity constraints under heavy-traffic scaling.

They define a corresponding prelimit formulation by imposing this bounded-control structure directly on the scaled queueing network while retaining the scaled state equation, admissibility constraints, and discounted cost objective. If the conjecture holds, the modified Brownian control problem would provide a rigorous heavy-traffic approximation for the original network control problem, thereby supporting the paper’s diffusion-based computational policy methodology.