Improved asymptotic bounds on critical transmission radius for greedy forward routing in wireless ad hoc networks
Li-Xin Wang, Chih Wei Yi, Frances Yao · 2008
Consider a random wireless ad hoc network represented by a Poisson point process over a unit-area disk with mean n. Let σn denote its critical transmission radius for greedy forward routing, and βo = 1/ (2/3 -- √3/2π) ≈ 1.62. It was recently proved that for any constant ε > 0, it is asymptotically almost sure that (1 -- ε) √βo in n/πn ≤ σn ≤ (1 + ε) √βo in n/πn. In this paper, we obtain tighter asymptotic bounds on σn. Specifically, we prove that for any constant c, the asymptotic probability of σn ≤ √βo in n + c/πn is at least 1 -- ( 1/1/βo--1/3 -- βo/2) e -c and at most e -βo/2 e -- c. Consequently, for any positive sequence (εn : n ≥ 1) with εn = o(In n) and εn → ∞, it is asymptotically almost sure that √βo in n --εn/πn ≤ σn ≤ √βo in n + εn/πn. We also conjecture that for any constant c, the asymptotic probability of σn ≤ √βo 1n n + c/σn is exactly exp (--(1/1/βo--1/3 -- β;o/2) e --c).