On the optimality of field-line routing in massively dense wireless multi-hop networks

We consider the load balancing problem in large wireless multi-hop networks by applying the continuum approximation. The task is to find routes, geometric curves, such that the maximal traffic load in the network is minimized. In finite fixed networks, multi-path routes generally yield a lower congestion and thus allow higher throughput. In contrast, we show that in dense wireless multi-hop networks, the optimal load balancing can be achieved by a destination based single-path routing referred to as field-line routing. This is because any routing can be transformed to the corresponding field-line routing with the same or better performance, by using as paths, the field lines of the so-called destination flow associated with the original routing. The concepts are illustrated with two examples. In the case of a unit disk with unit traffic, the maximal load of 0.389 of a multi-path routing system is reduced to 0.343 by using the field-line routing. Similar improvements are also demonstrated for the unit square.