# Simple Approximations of the SIR Meta Distribution in General Cellular Networks

## TL;DR

This paper introduces simple approximation methods for the SIR meta distribution in heterogeneous cellular networks, enabling easier analysis of network performance beyond standard success probability, especially for non-Poisson models.

## Contribution

It proposes a novel approximation approach for the SIR meta distribution in general HCNs using the ASAPPP method and characterizes the asymptotic gap with a simple parameter.

## Key findings

- The approximations perform well across various network models.
- The asymptotic gap $G_0$ effectively captures the difference between Poisson and non-Poisson models.
- The methods simplify the analysis of complex cellular network models.

## Abstract

Compared to the standard success (coverage) probability, the meta distribution of the signal-to-interference ratio (SIR) provides much more fine-grained information about the network performance. We consider general heterogeneous cellular networks (HCNs) with base station tiers modeled by arbitrary stationary and ergodic non-Poisson point processes. The exact analysis of non-Poisson network models is notoriously difficult, even in terms of the standard success probability, let alone the meta distribution. Hence we propose a simple approach to approximate the SIR meta distribution for non-Poisson networks based on the ASAPPP ("approximate SIR analysis based on the Poisson point process") method. We prove that the asymptotic horizontal gap $G_0$ between its standard success probability and that for the Poisson point process exactly characterizes the gap between the $b$th moment of the conditional success probability, as the SIR threshold goes to $0$. The gap $G_0$ allows two simple approximations of the meta distribution for general HCNs: 1) the per-tier approximation by applying the shift $G_0$ to each tier and 2) the effective gain approximation by directly shifting the meta distribution for the homogeneous independent Poisson network. Given the generality of the model considered and the fine-grained nature of the meta distribution, these approximations work surprisingly well.