  We investigate a probabilistic model for routeing of messages in
relay-augmented multihop ad-hoc networks, where each transmitter sends one
message to the origin. Given the (random) transmitter locations, we weight the
family of random, uniformly distributed message trajectories by an exponential
probability weight, favouring trajectories with low interference (measured in
terms of signal-to-interference ratio) and trajectory families with little
congestion (measured in terms of the number of pairs of hops using the same
relay). Under the resulting Gibbs measure, the system targets the best
compromise between entropy, interference and congestion for a common welfare,
instead of an optimization of the individual trajectories.
  In the limit of high spatial density of users, we describe the totality of
all the message trajectories in terms of empirical measures. Employing large
deviations arguments, we derive a characteristic variational formula for the
limiting free energy and analyse the minimizer(s) of the formula, which
describe the most likely shapes of the trajectory flow. The empirical measures
of the message trajectories well describe the interference, but not the
congestion; the latter requires introducing an additional empirical measure.
Our results remain valid under replacing the two penalization terms by more
general functionals of these two empirical measures.
