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In this paper, the authors consider a three node, two-way relay system with digital network coding. The aim is to minimize total energy consumption while ensuring queue stability at all nodes, for a given pair of random packet arrival rates. Specifically, they allow for a set of transmission modes and solve for the optimal fraction of resources allocated to each mode. First, they formulate and solve the static-channel problem, where all link gains are constant over the duration of transmission. Then they solve the fading-channel problem, where link gains are random. They call the latter the ergodic energy efficiency problem, and show that its solution has a water-filling structure.
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