  Results for Gaussian relay channels typically focus on maximizing
transmission rates for given locations of the source, relay and destination. We
introduce an alternative perspective, where the objective is maximizing
coverage for a given rate. The new objective captures the problem of how to
deploy relays to provide a given level of service to a particular geographic
area, where the relay locations become a design parameter that can be
optimized. We evaluate the decode and forward (DF) and compress and forward
(CF) strategies for the relay channel with respect to the new objective of
maximizing coverage. When the objective is maximizing rate, different locations
of the destination favor different strategies. When the objective is coverage
for a given rate, and the relay is able to decode, DF is uniformly superior in
that it provides coverage at any point served by CF. When the channel model is
modified to include random fading, we show that the monotone ordering of
coverage regions is not always maintained. While the coverage provided by DF is
sensitive to changes in the location of the relay and the path loss exponent,
CF exhibits a more graceful degradation with respect to such changes. The
techniques used to approximate coverage regions are new and may be of
independent interest.
