  Meticulous modelling and performance analysis of Low-Power Wide-Area (LPWA)
networks are essential for large scale dense Internet-of-Things (IoT)
deployments. As Long Range (LoRa) is currently one of the most prominent LPWA
technologies, we propose in this paper a stochastic-geometry-based framework to
analyse the uplink transmission performance of a multi-gateway LoRa network
modelled by a Matern Cluster Process (MCP). The proposed model is first to
consider all together the multi-cell topology, imperfect spreading factor (SF)
orthogonality, random start times, and geometric data arrival rates. Accounting
for all of these factors, we initially develop the SF-dependent collision
overlap time function for any start time distribution. Then, we analyse the
Laplace transforms of intra-cluster and inter-cluster interference, and
formulate the uplink transmission success probability. Through simulation
results, we highlight the vulnerability of each SF to interference, illustrate
the impact of parameters such as the network density, and the power allocation
scheme on the network performance. Uniquely, our results shed light on when it
is better to activate adaptive power mechanisms, as we show that an SF-based
power allocation that approximates LoRa ADR, negatively impacts nodes near the
cluster head. Moreover, we show that the interfering SFs degrading the
performance the most depend on the decoding threshold range and the power
allocation scheme.
