corresponds to the possible achievable rate at the sink node due to direct coherent transmission from the source and the relay and due to the overheard transmission from the source to the relay. The higher the channel attenuation, the less will be the contribution of farther nodes, “overheard” transmissions become less relevant, and coherent transmission reduces to a simple transmission from the previous relay. The system is then closer to simple store-and-forward relaying.

The authors of [@xie-kumar04network-information-theory-scaling-law] have shown that any rate strictly less than $R$ is achievable through the coding and decoding scheme. This achievable rate formula can also be obtained from the capacity formula of a physically degraded multi-relay channel (see [@reznik-etal04degraded-gaussian-multirelay-channel]), since the capacity of the degraded relay channel is a lower bound to the actual channel capacity. [*In this paper, we will seek to optimize $R$ in (\[eqn:achievable\_rate\_multirelay\]) over power allocations to the nodes and the node locations, keeping in mind that $R$ is a lower bound to the actual capacity. We denote the value of $R$ optimized over power allocation and relay locations by $R^*$.*]{}

Path-Loss Model
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We model the power gain via the exponential path-loss model: the power gain at a distance $r$ 