  The system-level performance of multi-gateway downlink long-range (LoRa)
networks is investigated in the present paper.
  Specifically, we first compute the active probability of a channel and the
selection probability of an active end-device (ED) in the closed-form
expressions. We then derive the coverage probability (Pcov) and the area
spectral efficiency (ASE) under the impact of the capture effects and different
spreading factor (SF) allocation schemes.
  Our findings show that both the Pcov and the ASE of the considered networks
can be enhanced significantly by increasing both the duty cycle and the
transmit power.
  Finally, Monte-Carlo simulations are provided to verify the accuracy of the
proposed mathematical frameworks.
