Probability distributions for the number of radio transceivers in a hot spot with an application to the evaluation of blocking probabilities

This paper considers two sets of points which can communicate with one another and which are uniformly and randomly distributed, one set within a confined area, a hot spot, on the plane, and the other with a different density over the rest of the plane: an inverse power law with lognormal shadowing is assumed for attenuation. Communication is assumed to be possible if the attenuation of a signal transmitted from one point, say a mobile telephone, does not exceed some specified value by the time it reaches the other, say a base station. The probability distribution of the number of base stations which can hear such a signal from a given mobile is found, taking account of the effect of the hot spot. The conditional distribution of the number of such base stations within the hot spot, given the total number over the whole plane, is deduced, and this result is applied in an example to finding the blocking probability for a directed retry algorithm. The results also apply to the number of mobiles within range of a base station and, in an ad hoc network, to the number of nodes within range of a given node.