Probability distributions for the number of radio transceivers which can communicate with one another

This paper considers points which can communicate with one another and which are uniformly and randomly distributed on the plane, assuming an inverse power law for attenuation and log-normal shadowing. First, a probability distribution is obtained for the distance between a pair of such points. Communication is assumed to be possible if the attenuation of a signal transmitted from one point, say a mobile phone, 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. An example shows how the probability that a mobile is in a handover region between two or more base stations can be evaluated. The analysis is extended to the case where the power law changes at a specified distance from the mobile, and also to consider the effect of an area within the plane containing a greater density of mobiles and/or base stations (a hot spot). A final result gives the conditional distribution for the number of base stations within some specified area of the plane, given the number within some larger area containing that specified area. 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.