Optimal and Approximately Optimal Control Policies for Queues in Heavy Traffic

摘要

We treat the 'approximately' optimal control problem for tandem queueing or production networks (with local feedback allowed) under heavy traffic. The buffers (scaled with traffic) are finite. The controls allow various inputs, connecting links and the processors to be shut down or opened, in order to manage the system. The service and arrival rates, as well as the routing probabilities can also be controlled, and the system statistics can depend on the system state (scaled buffer occupancies). The associated costs involve holding costs, costs for shutting off/on the links or processors and the opportunity cost for lost production. It is shown that the (scaled) controlled system converges weakly (in an appropriate sense) to a controlled limit 'reflected' diffusion. In the rescaled time, the actions of the controllers lead to multiple 'simultaneous' impulses in the limit problem. Thus we have a non-standard limit control problem, and the usual methods of weak convergence for systems under heavy traffic must be modified. Since the optical or nearly optimal controls for the physical process are usually not possible to get, it is of considerable interest to know whether an optimal or nearly optimal control for the limit process is also nearly optimal for the physical system with heavy traffic. This is shown to be true, under reasonable conditions. Although the limit control problem is non-standard and there is little available theory concerning it, acceptable numerical procedures are available.