Let the fixed power expended in a relay (for reception and driving the electronic circuits) be denoted by $P_{\mathrm{rcv}}$ (expressed in mW), and the initial battery energy in each node be denoted by $E$ (in mJ unit). The information theoretic approach utilized in this paper requires that the nodes in the network are always on. Hence, the lifetime of node $i, 1 \leq i \leq N$, is $\tau_i=\frac{E}{P_i+P_{rcv}×}$, the lifetime of the source is $\tau_0=\frac{E}{P_0}$, and that of the sink is $\tau_{N+1}=\frac{E}{P_{rcv}}$. The rate of battery replacement at node $i$ is $\frac{1}{\tau_i}$. Hence, the rate at which we have to replace the batteries in the network is $\sum_{i=0}^{N+1}\frac{1}{\tau_i×}=\frac{1}{E×}(\sum_{i=0}^{N}P_i+(N+1)P_{rcv})$. The depletion rate $\frac{P_{rcv}}{E}$ is inevitable at any node, and it does not affect the achievable data rate. Hence, in order to reduce the battery replacement rate, we must reduce the sum transmit power in the entire network.

Placement on a Line of Known Length {#sec:total_power_constraint}
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As a precursor to addressing the deploy-as-you-go problem over a line of unknown length, in this section we solve the problem of power constrained deployment of [*a given number of relays on a line of known length*]{}. We will often refer to this problem as 