Trading rate for balanced queue lengths for network delay minimization

We consider a communication channel with two transmitters and one receiver, with an underlying rate region which is approximated as a general pentagon. Different from the Gaussian multiple access channel (MAC) capacity region, the sum-rate on the dominant face of this pentagon is not a constant. We allocate rates from this rate region to users according to their current queue lengths in order to minimize the average delay in the system. We formulate the problem as a Markov decision problem (MDP), and derive the structural properties of the corresponding discounted-cost MDP. We show that the delayoptimal policy has a switch curve structure. For the discountedcost problem, we prove that the switch curve has a limit along one of the dimensions. The delay-optimal policy divides the entire queue state space into two via a switch curve. If the queue state is on one side of the switch curve, the system operates at one of the corner points of the rate pentagon which favors maximum sum-rate. When the queue state switches to the other side of the switch curve, the system operates at the other corner point of the rate pentagon which favors balancing the queue lengths. As a result, the system does not always operate at the sumrate maximizing rate pair, but trades rate for balanced queue lengths for the goal of minimizing the overall delay. The existence of a limit in the switch curve along one of dimensions implies that, once the queue state is beyond the limit, the system always operates at one of the corner points, implying that the queues can be operated partially distributedly.