Routeing Properties in a Gibbsian Model for Highly Dense Multihop Networks

• Published in: IEEE transactions on information theory 2019-11, Vol.65 (11), p.6875-6897
• Main Authors: Konig, Wolfgang, Tobias, Andras
• Format: Article
• Language: English
• Subjects: Ad hoc networks; Area measurement; Base stations; Entropy; expected number of hops; Gibbs distribution; high-density limit; Interference; message routeing; Multihop ad-hoc network; point processes; Probabilistic models; Probability theory; Qualitative analysis; Radio equipment; Relays; Routeing; selfish routeing; signal-to-interference ratio; Spread spectrum communication; Trajectory; Trajectory measurement; variational analysis
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2924187

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 other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution, which favors trajectories with low interference, measured in terms of signal-to-interference ratio. This model was introduced in our earlier paper, where we expressed, in the limit of a high density of users, the typical distribution of the family of trajectories in terms of a law of large numbers. In this paper, we derive its qualitative properties. We analytically identify the emerging typical scenarios 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, regime 1 in the limit of a large communication area and large distances and regime 2 in the limit of a strong interference weight. In both the regimes, the typical trajectory approaches a straight line quickly, in regime 1 with equal hop lengths. Interestingly, in regime 2, the typical length of a hop diverges logarithmically in the distance of the transmitter to the base station. We further analyze (regime 3) local and global repulsive effects of a densely populated subarea on the trajectories.