A Geometric Theorem for Wireless Network Design Optimization

Consider an infinite square grid . How many discs of given radius , centered at the vertices of , are required, in the worst case, to completely cover an arbitrary disc of radius placed on the plane? We show that this number is an integer in the set whose value depends on the ratio of to the grid spacing. This result can be applied at the very early design stage of a wireless cellular network to determine, under the recent International Telecommunication Union (ITU) proposal for a traffic load model, and under the assumption that each client is able to communicate if it is within a certain range from a base station, conditions for which a grid network design is cost effective, for any expected traffic demand.