Proof of convergence of LoRaWAN model

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.