  LoRa networks are pivotally enabling Long Range connectivity to low-cost and
power-constrained user equipments (UEs) in a wide area, whereas a critical
issue is to effectively allocate wireless resources to support potentially
massive UEs while resolving the prominent near-far fairness issue, which is
challenging due to the lack of tractable analytical model and the practical
requirement for low-complexity and low-overhead design. Leveraging on
stochastic geometry, especially the Poisson rain model, we derive (semi-)
closed-form formulas for the aggregate interference distribution, packet
success probability and hence system throughput in both single-cell and
multi-cell setups with frequency reuse, by accounting for channel fading,
random UE distribution, partial packet overlapping, and/or multi-gateway packet
reception. The analytical formulas require only average channel statistics and
spatial UE distribution, which enable tractable network performance evaluation
and incubate our proposed Iterative Balancing (IB) method that quickly yields
high-level policies of joint spreading factor (SF) allocation, power control,
and duty cycle adjustment for gauging the average max-min UE throughput or
supported UE density with rate requirements. Numerical results validate the
analytical formulas and the effectiveness of our proposed optimization scheme,
which greatly alleviates the near-far fairness issue and reduces the spatial
power consumption, while significantly improving the cell-edge throughput as
well as the spatial (sum) throughput for the majority of UEs, by adapting to
the UE/gateway densities.
