Theorem \[theorem:single\_relay\_total\_power\]) $\lim_{\lambda \rightarrow \infty} \frac{y_{1}^{*}}{L}=\frac{1}{2×}$, $\lim_{\lambda \rightarrow \infty}P_{0,2}=0$ and $\lim_{\lambda \rightarrow 0}P_{0,1}=P_{T}$. For large values of $\lambda$, source and relay cooperation provides negligible benefit since source to sink attenuation is very high. So it is optimal to place the relay at a distance $\frac{L}{2×}$. The relay works as a repeater which forwards data received from the source to the sink. For small $\lambda$, the gain obtained from coherent transmission is dominant, and, in order to receive sufficient information (required for coherent transmission) from the source, the relay is placed near the source.

Optimal Placement for a Multi-Relay Channel {#subsection:properties_of_gain}
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As we discussed earlier, we need to place $N$ relay nodes such that $\frac{1}{g_{0,1}×}+\sum_{k=2}^{N+1} \frac{(g_{0,k-1}-g_{0,k})}{g_{0,k}g_{0,k-1}\sum_{l=0}^{k-1}\frac{1}{g_{0,l}×}×}$ is minimized. Here $g_{0,k}=e^{-\rho y_{k}}$. We have the constraint $0 \leq y_{1} \leq y_{2} \leq \cdots \leq y_{N} \leq y_{N+1}=L$. Now, writing $z_{k}=e^{\rho y_{k}}$, and defining $z_{0}:=1$, we arrive at the following problem: $$\begin{aligned}
& & \min \bigg\{ z_{1}+\sum_{k=2}^{N+1} \frac{z_{k}-z_{k-1}}{\sum_{l=0}^{k-1} z_{l}}\bigg\}\nonumber\\
& s.t & \,\, 1 \leq z_{1} \leq \cdots \leq z_{N} \leq z_{N+1} = e^{\lambda} \label{eqn:multirelay_optimization}\end{aligned}$$ The objective function is convex in each of the variables $z_{1}, z_{2},\cdots, z_{N}$. The objective function is sum of linear fractionals, and the constraints are linear. [*Remark:*]{} 