Even one-dimensional mobility increases ad hoc wireless capacity

The study of the capacity of wireless ad hoc networks has received significant attention. Gupta and Kumar (see IEEE, Transactions on Information Theory, vol.46, no.2, p.388-404, 2000) considered a model in which n nodes are randomly located in a disk of unit area and each node has a random destination node it wants to communicates to. They showed that as the number of nodes n increases, the throughput per source and destination (S-D) pair goes to zero like 1/√n even allowing for optimal scheduling and relaying of packets. The nodes are however assumed to be fixed. Grossglauser and Tse (see IEEE INFOCOM, Anchorage, Alaska, p.1360-1369, 2001) considered an alternative model in which the nodes are mobile, and they showed that in sharp contrast to the fixed node case, the throughput per S-D pair can actually be kept constant even as the number of nodes scales. This performance gain is obtained through a multiuser diversity effect. In the mobility model considered by Grossglauser and Tse, the trajectory of each node i is an independent, stationary and ergodic random process X_i(t) with a uniform stationary distribution on the unit disk. Intuitively, this implies that a sample path of each node "fills the space over time". This mobility model is unrealistic in many practical settings. A natural question that arises is then how strongly the throughput result of Grossglauser et al. depends on this mobility model. We show in this paper that the throughput result of Grossglauser et al. still holds even when nodes have much more limited mobility patterns. Specifically, we consider a model in which each node i is constrained to move on a single-dimensional great circle G_i on the unit sphere. Each node moves randomly along its own circle. The throughput capacity of such a network of course depends on the configuration of the great circles. Our main result is that if the locations of the great circles are chosen randomly and independently, then for almost all configurations of such great circles, the throughput per S-D pair can be kept constant as the number of nodes increases. Thus, although each node is restricted to move in a one-dimensional space, the same asymptotic performance is achieved as in the case when they can move in the entire 2-D region- .