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Analysis of SINR Outage in Large-Scale Cellular Networks Using Campbell's Theorem and Cumulant Generating Functions

The signal-to-noise-plus-interference ratio (SINR) outage probability is one of the key performance parameters of a wireless cellular network, and its analytical as well as numerical evaluation has occupied many researchers. Recently, the introduction of stochastic geometric modeling of cellular networks has brought the outage problem to the forefront again. A popular and powerful approach is to exploit the available moment generating function (or Laplace transform) of received signal and interference, whenever it exists, by applying the Gil-Pelaez inversion formula. However, with the stochastic geometric modeling, the moment generating function may either be too complicated to exist in closed-form or at worst may not exist. Toward this end, in this paper, we study two alternate ways of evaluating the SINR outage. In the first case, we emphasize the significance of calculating cumulants over moments and exploit the fact that the cumulants of point processes are easily calculable using Campbell's theorem. The SINR outage is then analytically characterized by Charlier expansion based on Gaussian and Student's $t$-distributions and their associated Hermite and Krishnamoorthy polynomials. In the second case, we exploit the saddle point method, which gives a semi-analytical method of calculating the SINR outage, whenever the cumulant generating function of received signal and interference exists. For the purpose of demonstration, we apply these techniques on a downlink cellular network model where a typical user experiences a coordinated multi-point transmission, and the base stations are modeled by homogeneous Poisson point process. For the convenience of readers, we also provide a brief overview of moments, cumulants, their generating functions, and Campbell's theorem, without invoking measure theory. Numerical results illustrate the accuracy of the proposed mathematical approaches.