Asymptotic Critical Transmission Radii in Wireless Networks over a Convex Region

Critical transmission ranges (or radii) in wireless ad-hoc and sensor networks have been extensively investigated for various performance metrics such as connectivity, coverage, power assignment and energy consumption. However, the regions on which the networks are distributed are typically either squares or disks in existing works, which seriously limits the usage in real-life applications. In this article, we consider a convex region (i.e., a generalisation of squares and disks) on which wireless nodes are uniformly distributed. We have investigated two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, respectively, and have also established their precise asymptotic distributions. These make the previous results obtained under the circumstance of squares or disks special cases of this work. More importantly, our results reveal how the region shape impacts on the critical transmission ranges: it is the length of the boundary of the (fixed-area) region that completely determines the transmission ranges. Furthermore, by isodiametric inequality, the smallest critical transmission ranges are achieved when regions are disks only.