Internets in the Sky: The Capacity of Three Dimensional Wireless Networks

Consider n nodes located in a sphere of volume V cubic meters, each capable of transmitting at a rate of W bits/sec. Under a protocol based model for successful receptions, the entire network can carry only theta (WV 1 over 3 n 2 over 3) bit-meters/sec, where 1 bit carried a distance of 1 meter is counted as 1 bit-meter. This is the best possible even assuming the node locations, traffic patterns, and the range/power/ timing of each transmission, are all optimally chosen. If the node locations and their destinations are randomly chosen, and all transmissions employ the same power/range, then each node only obtains a throughput of theta (W over (n log sup 2 n)sup (1 over 3)) bits/sec, if the network is optimally operated. Similar results hold under an alternate physical model where a minimun signal-to-interference ratio is specified for successful receptions. The proofs of these results require determination of the VC-dimensions of certain geometric sets, which may be of independent interest.