  We propose analytical models that allow us to investigate the performance of
long range wide area network (LoRaWAN) uplink in terms of latency, collision
rate, and throughput under the constraints of the regulatory duty cycling, when
assuming exponential inter-arrival times. Our models take into account sub-band
selection and the case of sub-band combining. Our numerical evaluations
consider specifically the European ISM band, but the analysis is applicable to
any coherent band. Protocol simulations are used to validate the proposed
models. We find that sub-band selection and combining have a large effect on
the quality of service (QoS) experienced in an LoRaWAN cell for a given load.
The proposed models allow for the optimization of resource allocation within a
cell given a set of QoS requirements and a traffic model.
