A Gibbsian model for message routing in highly dense wireless networks

In spatial telecommunication networks, it is a prominent question how to route many messages in the same time. We propose a random mechanism for routing messages in a network, where users are situated according to a Poisson point process in a compact subset of $\mathbb R^d$, and each user sends one message to the single base station. Messages are transmitted either directly or via other users, with a given upper bound on the number of hops. Given the locations of users, we define a Gibbs distribution on the

set of all such trajectory families, which favours trajectories with little interference (measured in terms of the signal-to-interference ratio (SIR)) and trajectory families with little congestion (measured in terms of the number of pairs of incoming messages of the users).

We derive the behaviour of this system in the limit of a high spatial density of users using large-deviation methods, compute the limiting free energy and provide a law of large numbers for the empirical measure of message trajectories. The limit of these empirical measures is given as the minimizer(s) of a characteristic variational formula. In the special case when congestion is not penalized, we analyze the minimizer and investigate the questions of the typical number of hops, the typical length of a hop and the typical shape of a trajectory in the highly dense network.

The topic of this talk is joint work with Wolfgang König.