Performance of 3-Dimensional Scans with Narrow-Beam Directional Antennas

We present an analytical technique to model the performance of any sequence of scan directions over any search space. Intuitively, sequences that cover the search space in the minimum time are preferred. However, we find that the important factors in measuring its performance are not only (1) the number of points required to cover the space, but also (2) the degree of overlap between different scan points, and (3) the probability of success of any particular detection attempt. Note that (1) and (2) are inevitably correlated. For scans that maintain a constant transmit power and antenna gain, increasing the number of aimpoints, and the overlap, has a negligible effect on the mean detection time. However, we argue that using variable transmission power results in better network topologies by emphasizing the detection of nearby nodes. In this case, shorter sequences detect long- range nodes more quickly. This difference does not make it "better," but merely increases the relative population of long- range links. The favoritism shown to near nodes can be easily and precisely controlled through sequence design. Our findings provide useful tools and insights for the protocol designer.