improved asymptotic bounds on critical transmission radius for greedy forward routing in wireless ad hoc networks
Consider a random wireless ad hoc network represented by a Poisson point process over a unit-area disk with mean n . Let ? n denote its critical transmission radius for greedy forward routing, and ? o = 1/ (2/3 -- ?3/2?) ? 1.62. It was recently proved that for any constant ? > 0, it is asymptotically almost sure that (1 -- ?) ?? o in n /? n ? ? n ? (1 + ?) ?? o in n /? n . In this paper, we obtain tighter asymptotic bounds on ? n . Specifically, we prove that for any constant c , the asymptotic probability of ? n ? ?? o in n + c /? n is at least 1 -- ( 1/1/? o --1/3 -- ? o /2) e - c and at most e -? o /2 e -- c . Consequently, for any positive sequence (? n : n ? 1) with ? n = o (In n ) and ? n ? ?, it is asymptotically almost sure that ?? o in n --? n /? n ? ? n ? ?? o in n + ? n /? n . We also conjecture that for any constant c , the asymptotic probability of ? n ? ?? o 1n n + c /? n is exactly exp (--(1/1/? o --1/3 -- ?; o /2) e -- c ).
greedy forward routing wireless ad hoc networks random deployment