Routeing properties in a Gibbsian model for highly dense multihop networks
- König, Wolfgang
- Tóbiás, András
2010 Mathematics Subject Classification
- 60G55 60K30 65K10 82B21 90B15 90B18 91A06
- Multihop ad-hoc network, signal-to-interference ratio, Gibbs distribution, message routeing, high-density limit, point processes, variational analysis, expected number of hops, expected length of a hop, deviation from the straight line, selfish optimization
We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signal-to-interference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the high-density limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equally-sized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a game-theoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized
- IEEE Trans. Inform. Theory, 65 (2019), pp. 6875--6897, DOI 10.1109/TIT.2019.2924187 .