[*offline deployment problem*]{}. The results of this section provide (i) first insights into the relay placements we obtain using the multi-relay channel model, (ii) a starting point for the formulation of as-you-go deployment problem, and (iii) a benchmark with which we can compare the performance of our as-you-go deployment algorithm.

Optimal Power Allocation {#subsec:optimal_power_allocation}
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In this section, we consider the optimal placement of relay nodes on a line of given length, $L$, so as to to maximize $R$ (see (\[eqn:achievable\_rate\_multirelay\])), subject to a total power constraint on the source and relay nodes given by $\sum_{i=0}^{N}P_{i}=P_{T}$. We will first maximize $R$ in (\[eqn:achievable\_rate\_multirelay\]) over $P_{i,j}, 0 \leq i < j \leq (N+1)$ for any given placement of nodes (i.e., given $y_{1}, y_{2},\cdots,y_{N}$). This will provide an expression of achievable rate in terms of channel gains, which has to be maximized over $y_{1}, y_{2},\cdots,y_{N}$. Let $\gamma_{k}:=\sum_{i=0}^{k-1}P_{i,k}$ for $k \in \{1,2,\cdots,N+1\}$ (expressed in mW). Hence, the sum power constraint becomes $\sum_{k=1}^{N+1}\gamma_{k}=P_{T}$.

\[theorem:multirelay\_capacity\]

1.  Under the exponential path-loss model, for fixed location of relay nodes, the optimal power allocation that maximizes the achievable rate for the sum power constraint is given by: $$P_{i,j}=
    \begin{cases}
    \frac{g_{i,j}}{\sum_{l=0}^{j-1}g_{l,j}×}\gamma_{j}\,\, & \forall 0 \leq i <j \leq (N+1) 