  We use information theoretic achievable rate formulas for the multi-relay
channel to study the problem of as-you-go deployment of relay nodes. The
achievable rate formulas are for full-duplex radios at the relays and for
decode-and-forward relaying. Deployment is done along the straight line joining
a source node and a sink node at an unknown distance from the source. The
problem is for a deployment agent to walk from the source to the sink,
deploying relays as he walks, given that the distance to the sink is
exponentially distributed with known mean. As a precursor, we apply the
multi-relay channel achievable rate formula to obtain the optimal power
allocation to relays placed along a line, at fixed locations. This permits us
to obtain the optimal placement of a given number of nodes when the distance
between the source and sink is given. Numerical work suggests that, at low
attenuation, the relays are mostly clustered near the source in order to be
able to cooperate, whereas at high attenuation they are uniformly placed and
work as repeaters. We also prove that the effect of path-loss can be entirely
mitigated if a large enough number of relays are placed uniformly between the
source and the sink. The structure of the optimal power allocation for a given
placement of the nodes, then motivates us to formulate the problem of as-you-go
placement of relays along a line of exponentially distributed length, and with
the exponential path-loss model, so as to minimize a cost function that is
additive over hops. The hop cost trades off a capacity limiting term, motivated
from the optimal power allocation solution, against the cost of adding a relay
node. We formulate the problem as a total cost Markov decision process,
establish results for the value function, and provide insights into the
placement policy and the performance of the deployed network via numerical
exploration.
