Improved Asymptotic Bounds on Critical Transmission Radius for Greedy Forward Routing in Wireless Ad Hoc Networks

Abstract

Consider a random wireless ad hoc network represented by a Poisson point process over a unit-area disk with mean n. Let sigma(n) denote its critical transmission radius for greedy forward routing, and beta(0) = 1/(2/3 - root 3/2 pi) approximate to 1.6(2). It was recently proved that for any constant epsilon > 0, it is asymptotically almost sure that (1 - epsilon) root beta(0) 1n n/pi n <= sigma(n) <= (1 + epsilon) root beta(0) 1n n/pi n. In this paper, we obtain tighter asymptotic bounds on sigma(n). Specifically, we prove that for any constant c, the asymptotic probability of sigma(n) <= root beta(0) ln n+c/pi n is at least 1-(1/1/beta(0)-1/3-beta(0)/2)e(-c) and at most e(-beta 0/2e-c) Consequently, for any positive sequence (xi(n) : n >= 1) with xi(n) = o(ln n) and xi(n) -> proportional to, it is asymptotically almost sure that root beta(0) ln n-xi(n)/pi n <= sigma(n) <= root beta(0) ln n+xi(n)/pi n We also conjecture that for any constant c, the asymptotic probability of sigma(n) <= root beta(0) ln n+c/pi n is exactly exp (-(1/1/beta(0)-1/3 - beta(0)/2)e(-c)).