The Critical Grid Size and Transmission Radius for Local-Minimum-Free Grid Routing in Wireless Ad Hoc and Sensor Networks

Abstract

In grid routing, the plane is tessellated into equal-sized square cells. Two cells are called neighbor cells if they share a common edge, and two nodes are called routing neighbors if they are in neighbor cells and within each other's transmission range. If communication parties are in the same cell, packets can be transmitted directly; otherwise, packets are forwarded to routing neighbors that are in cells closer to destination cells. As a greedy strategy, grid routing suffers the existence of local minima at which no neighbor nodes exist for relaying packets. To guarantee deliverability, in this paper, we investigate two vital parameters of grid routing, called the grid size and the transmission radius. Assume that nodes are represented by a Poisson point process with rate n over a unit-area square, and let l denote the grid size and r the transmission radius. First, we show that if l = root beta ln n/n for some constant beta and r = root 5l, then beta = 1 is the threshold for deliverability. In other words, there almost surely do not exist local minima if beta > 1 and there almost surely exist local minima if beta < 1. Next, for any given beta > 1, we give sufficient and necessary conditions to determine the critical transmission radius (CTR) for deliverability. Then, we show that as beta congruent to 1.092, the CTR r congruent to 2.09 root ln n/nis the minimum over all beta > 1. Simulation results are given to validate this theoretical work.