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

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 it over a unit-area square, and let I denote the grid size and r the transmission radius. First, we show that if I = [the square root of]βln n/n for some constant β and r = 5l, then β = 1 is the threshold for deliverability. In other words, there almost surely do not exist local minima if β > 1 and there almost surely exist local minima if β < 1. Next, for any given β > 1, we give sufficient and necessary conditions to determine the critical transmission radius (CTR) for deliverability. Then, we show that as β [congruent with] 1.092, the CTR r [congruent with] [the square root of] 2.09ln n/n is the minimum over all β > 1. Simulation results are given to validate this theoretical work. [PUBLICATION ABSTRACT]