From optimization problem (\[eqn:multirelay\_optimization\]) we observe that optimum $z_{1},z_{2},\cdots,z_{N}$ depend only on $\lambda:=\rho L$. Since $z_{k}=e^{\lambda \frac{y_{k}}{L}}$, the normalized optimal distance of relays from the source depend only on $\lambda$ and $N$.

\[theorem:capacity\_increasing\_with\_N\] For fixed $\rho$, $L$ and $\sigma^{2}$, the optimized achievable rate $R^{*}$ for a sum power constraint *strictly* increases with the number of relay nodes.

See Appendix \[appendix:proof\_of\_capacity\_increases\_in\_N\].

\[theorem:G\_increasing\_in\_lambda\] For any fixed number of relays $N \geq 1$, $G(N,\lambda)$ is increasing in $\lambda$.

See Appendix \[appendix:proof\_of\_G\_increasing\_in\_lambda\].

[**A numerical study of multi-relay placement:**]{} We discretize the interval $[0,L]$ and run a search program to find normalized optimal relay locations for different values of $\lambda$ and $N$. The results are summarized in Figure \[fig:multirelay-optimal-location\].

We observe that at low attenuation (small $\lambda$), relay nodes are clustered near the source node and are often at the source node, whereas at high attenuation (large $\lambda$) they are almost uniformly placed along the line. For large $\lambda$, the effect of long distance between any two adjacent nodes dominates the gain obtained by coherent relaying. Hence, it is beneficial to minimize the maximum distance between any two adjacent nodes and thus multihopping is a better strategy in this case. For small $\lambda$, the gain 