Matthias Ludewig (Adelaide)
András Tóbiás (TU Berlin)
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 distribution, 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 of the formula, which describes the most likely shape 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 with more general functionals of these two empirical measures. We also analyse qualitative properties of the minimizer in the special case when congestion is not penalized.