Proof of convergence of LoRaWAN model

TL;DR

This paper provides an analytical proof of convergence for a LoRaWAN network performance model, demonstrating that fixed-point iterations reliably reach a solution with few iterations needed.

Contribution

It offers the first rigorous proof of convergence for the LoRaWAN performance model and validates the practical efficiency of fixed-point iterative solutions.

Findings

Existence of a fixed point solution is proven analytically.

Fixed-point iterations converge rapidly in experiments.

Few iterations are sufficient for convergence.

Abstract

In this document, we prove the convergence of the model proposed in [1], which aims at estimating the LoRaWAN network performance in a single-gateway scenario. First, we provide an analytical proof of the existence of a fixed point solution for such a system. Then, we report experimental results, showing that the system of the two inter-dependent equations provided by the model can be solved through fixed-point iterations, and that a limited number of iterations is enough to reach convergence.