Energy optimization for stable two-way relaying with a multi-access uplink

In this paper, we consider a three-node, two-way relay system with digital network coding over static channels. For a given pair of random packet arrival rates, we aim to minimize total energy consumption while ensuring queue stability at all nodes. A set of transmission modes is considered and we solve for the optimal fraction of resources allocated to each mode, including a multi-access uplink transmission mode and a network coded broadcasting mode. For the downlink, we further consider whether it is more energy-efficient, to superimpose the excess bits of the larger message for one user with the network coded message for both users. We formulate and solve the corresponding optimization problem, deriving conditions under which it is better to use superposition coding. Finally, we present a detailed analysis of the queues at each node using a random scheduling method that closely approximates the theoretical design, through a Markov chain model.