line of unknown random length. Numerical work on as-you-go deployment has been presented in Section \[section:numerical\_work\_information\_theoretic\_model\]. Some discussions are provided in Section \[section:additional\_discussion\]. Conclusions are drawn in Section \[conclusion\].

System Model and Notation {#sec:system_model_and_notation}
=========================

The Multi-Relay Channel
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The multi-relay channel was studied in [@xie-kumar04network-information-theory-scaling-law] and [@reznik-etal04degraded-gaussian-multirelay-channel] and is an extension of the single relay model presented in [@cover-gamal79capacity-relay-channel]. We consider a network deployed on a line with a source node and a sink node at the end of the line, and $N$ full-duplex relay nodes as shown in Figure \[fig:general\_line\_network\]. The relay nodes are numbered as $1, 2,\cdots,N$. The source and sink are indexed by $0$ and $N+1$, respectively. The distance of the $k$-th node [*from the source*]{} is denoted by $y_{k}:=r_{1}+r_{2}+\cdots+r_{k}$. Thus, $y_{N+1}=L$. As in [@xie-kumar04network-information-theory-scaling-law] and [@reznik-etal04degraded-gaussian-multirelay-channel], we consider the [*scalar, time-invariant, memoryless, AWGN setting.*]{}

We use the model that a symbol transmitted by node $i$ is received at node $j$ after multiplication by the (positive, real valued) channel gain $h_{i,j}$ (an assumption often made in the literature, see e.g., [@xie-kumar04network-information-theory-scaling-law] and [@gupta-kumar03capacity-wireless-networks]). The *power gain* from Node $i$ to Node $j$ is denoted by $g_{i,j} = h_{i,j}^2$. We define $g_{i,i}=1$ and $h_{i,i}=1$. The Gaussian additive 