Stochastic Geometry Analysis of Wireless Networks Using Matrix Laplace Transforms

In this paper, we consider a matrix function generalization of the Laplace transform of a random variable, termed the matrix Laplace transform. We characterize the conditions under which the matrix Laplace transform exists, establish its relation to the higher order moments and CCDF of a random variable, and derive the matrix Laplace transform of general Poisson shot noise. Techniques leveraging matrix Laplace transforms can provide improved tractability in the analysis of wireless networks using stochastic geometry. In particular, when one considers the underlying point process of transmitters in the network to follow a Poisson Point Process (PPP), techniques exploiting matrix Laplace transforms provide tractable expressions for the coverage probability of the network when the fading power on the desired signal follows a general phase-type distribution, the metadistribution of the SINR when the fading power on the desired signal follows an exponential distribution, and the distribution of the interference power observed by the typical user in the network.