k-coverage probability in a finite wireless network

We present a general mathematical framework to characterize the performance of an arbitrarily-located reference receiver in a finite wireless network. Modeling the locations of nodes as a uniform binomial point process (BPP), we derive the general k-coverage probability, which is the distribution of the signal-to-interference ratio (SIR) at the reference receiver when it connects to k(th) closest transmitting node. This k-coverage result for an arbitrarily-located reference receiver significantly generalizes existing works on finite networks, which usually assume that the reference receiver connects to a pre-selected transmitter located at a fixed distance that may not be a part of the BPP (ad hoc network setup). A particular special case of interest is that of k = 1, which can be interpreted as the downlink coverage probability in a finite cellular network modeled as a BPP. To the best of our understanding, the exact characterization of this result for finite cellular networks is not known. The mathematical analysis of k-coverage probability is enabled by the derivation of joint distance distributions from interfering and serving nodes to the reference receiver. As expected, our results demonstrate that the k-coverage probability strongly depends upon the location of the reference receiver. This observation highlights the importance of location information of the reference receiver for the accurate analysis of finite wireless networks.