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

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 WV 1 3n 2 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, traÆc 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 W (n log n) 1 3 bits/sec, if the network is optimally operated. Similar results hold under an alternate physical model where a minimum signal-tointerference ratio is speci ed for successful receptions. The proofs of these results require determination of the VC-dimensions of certain geometric sets, which may be of independent interest. This material is based upon work partially supported by the OÆce of Naval Research under Contract No. N00014-99-1-0696, the Air Force OÆce of Scienti c Research under Contract No. AF-DC-5-36128, by EPRI and the U.S. Army Research OÆce under Subcontract to Cornell University Contract Nos. WO8333-04 and Army OSP 35352-6086. Any opinions, ndings, and conclusions are those of the authors and do not necessarily re ect the views of the above agencies. Please address all correspondence to the second author at University of Illinois, Coordinated Science Laboratory, 1308 West Main Street, Urbana, IL 61801. Web http://black.csl.uiuc.edu/ prkumar