Even One - Dimensional Mobility Increases

I. Summary The study of the capacity of wireless ad hoc networks has received signiicant attention recently. Gupta and Kumar 1] 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= p n even allowing for optimal scheduling and relaying of packets. The nodes are however assumed to be xed. Grossglauser and Tse 2] considered an alternative model in which the nodes are mobile, and they showed that in sharp contrast to the xed 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 eeect. In the mobility model considered in 2], the trajectory of each node i is an independent, stationary and ergodic random process Xi(t) with a uniform stationary distribution on the unit disk. Intuitively, this implies that a sample path of each node \\lls 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 in 2] depends on this mobility model. We show in this paper that the throughput result in 2] still holds even when nodes have much more limited mobility patterns. Speciically, we consider a model in which each node i is constrained to move on a single-dimensional great circle Gi on the unit sphere. Each node moves randomly along its own circle. The throughput capacity of such a network of course depends on the connguration 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 conngura-tions 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. The constellation C = fG1; : : : ; Gng is a set of n great circles on the unit sphere. Each great circle Gi is randomly placed independently and uniformly on the sphere, but once generated, the …