Spatial distribution of traffic density in ALOHA with capture

We demonstrate how to calculate the packet traffic density as a function of distance from a central receiver in an ALOHA network with capture parameter α>1. Given a general function for the desired throughput density inside a circle of radius ρ and boundary conditions, the packet traffic density is derived at all radii less than ρ. The packet traffic density is first obtained in an annulus between radii ρ and ρ/α, and then is found in each of a number of successive inner annuli in terms of the solution in the adjacent outer annulus. Applications of this method are illustrated for a few throughput densities, boundary conditions, and capture parameters. Restriction of a class of throughput densities to a region of radius ρ is shown to be necessary in order to obtain a finite packet traffic density, and an upper bound on ρ is found in this case