The SINR Meta Distribution in Poisson Cellular Networks

Abstract

This work studies the signal-to-interference-plus-noise ratio (SINR) meta distribution (MD) in cellular networks with a focus on the Poisson model. Firstly, we show that for stationary base station point processes, arbitrary fading, and power-law path loss with exponent <inline-formula> <tex-math notation= LaTeX >$\\alpha $ </tex-math></inline-formula>, the base station density <inline-formula> <tex-math notation= LaTeX >$\\lambda $ </tex-math></inline-formula> and the noise power <inline-formula> <tex-math notation= LaTeX >$\\sigma ^{2}$ </tex-math></inline-formula> impact the SINR MD only through <inline-formula> <tex-math notation= LaTeX >$\\eta \\pmb {\\triangleq } \\lambda ^{\\alpha /2}/\\sigma ^{2}$ </tex-math></inline-formula>, termed the <italic>network signal-to-noise ratio</italic> (NSNR). Next, we show that for Poisson cellular networks, the SINR MD can be written as <inline-formula> <tex-math notation= LaTeX >$g(x)\\theta ^{-2/\\alpha }$ </tex-math></inline-formula> when the target SINR <inline-formula> <tex-math notation= LaTeX >$\\theta $ </tex-math></inline-formula> and the target reliability <inline-formula> <tex-math notation= LaTeX >${x}$ </tex-math></inline-formula> jointly satisfy a constraint. We derive this constraint and the integral of <inline-formula> <tex-math notation= LaTeX >$g(x)$ </tex-math></inline-formula>. Lastly, we discuss several extensions of the results to more general models and architectures.