LoRaWAN infrastructure has become widely deployed to provide wireless communications for various sensor applications. These applications generate different traffic volumes and require different quality of service (QoS). The paper presents an accurate mathematical model of low-power data transmission in a LoRaWAN sensor network, which allows accurate validation of key QoS indices, such as network capacity and packet loss ratio. Since LoRaWAN networks operate in the unlicensed spectrum, the model takes into account transmission attempt failures caused by random noise in the channel. Given QoS requirements, we can use the model to study how the performance of a LoRaWAN network depends on the traffic load and other scenario parameters. Since in LoRaWAN networks the transmissions at different modulation and coding schemes (MCSs) typically do not collide, we use the model to assign MCSs to the devices to satisfy their QoS requirements.