Energy harvesting diamond channel with energy cooperation

We consider the energy harvesting diamond channel, where the source and two relays harvest energy from nature, the relays help deliver the source´s messages via signal cooperation, and the source has the option of wirelessly transferring some of its energy to the relays via energy cooperation. We find the optimal offline transmit power allocations and energy transfer policies that maximize the end-to-end throughput. For the case of no energy cooperation, we decompose the problem into inner and outer maximization problems, and solve the overall problem iterating between the two. We show that the class of procrastinating policies, where energy is transferred only when it will be immediately used, is optimal. We then show that the problem with energy cooperation is equivalent to a problem without energy cooperation with suitably modified rate expressions. We show that, in this system, if the source sends more energy to a relay, then it sends less data, showing us how data and energy should flow together optimally in this network.