Separability, Asymptotics, and Applications of the SIR Meta Distribution in Cellular Networks

The signal-to-interference-ratio (SIR) meta distribution (MD) characterizes the link performance in interference-limited wireless networks: it evaluates the fraction of links that achieve an SIR threshold theta with a reliability above x. In this work, we show that in Poisson networks, for any independent fading and power-law path loss with exponent alpha, the SIR MD can be expressed as the product of theta(-2/alpha) and a function of x when (theta, x) is in the so-called "separable region". We show by simulation that the separable form serves as a good approximation of the SIR MD in Ginibre and triangular lattice networks when theta is chosen large enough. Given the quest for ultra-reliable transmission, we study the asymptotics of the SIR MD as x -> 1 for general cellular networks with Rayleigh fading. Finally, we apply our results to characterize the distribution of the link rate, where each link transmits with a rate satisfying a given reliability x, and the asymptotic distribution of the local delay, defined as the number of transmissions needed for a message to be received successfully.