Asymptotic Connectivity in Wireless Ad Hoc Networks Using Directional Antennas

Connectivity is a crucial issue in wireless ad hoc networks (WANETs). Gupta and Kumar have shown that in WANETs using omnidirectional antennas, the critical transmission range to achieve asymptotic connectivity is $O(sqrt{log n/n})$ if $n$ nodes are uniformly and independently distributed in a disk of unit area. In this paper, we investigate the connectivity problem when directional antennas are used. We first assume that each node in the network randomly beamforms in one beam direction. We find that there also exists a critical transmission range for a WANET to achieve asymptotic connectivity, which corresponds to a critical transmission power (CTP). Since CTP is dependent on the directional antenna pattern, the number of beams, and the propagation environment, we then formulate a non-linear programming problem to minimize the CTP. We show that when directional antennas use the optimal antenna pattern, the CTP in a WANET using directional antennas at both transmitter and receiver is smaller than that when either transmitter or receiver uses directional antenna and is further smaller than that when only omnidirectional antennas are used. Moreover, we revisit the connectivity problem assuming that two neighboring nodes using directional antennas can be guaranteed to beamform to each other to carry out the transmission. A smaller critical transmission range than that in the previous case is found, which implies smaller CTP.