Packet Collision Probability of Direct-to-Satellite IoT Systems

We investigate the packet collision probability among uncoordinated devices, in the uplink of DtS-IoT. Both the satellite spot shape and its motion along its orbital path are considered in the analysis. We analyze the probability of no uplink collision under two DtS-IoT settings: (i) unconfirmed ALOHA over a single channel, e.g., LoRaWAN class A with LoRa-CSS modulation; (ii) unconfirmed ALOHA with frequency hopping compliant with LoRaWAN class A with LR-FHSS. A closed-form solution is derived for the former, while an upper bound is found for the latter. The analytical results are validated by comparison with the outcomes of extensive simulations, showing that the obtained closed-form expressions accurately predict the probability of no collision. Moreover, the upper bound for the frequency hopping case is proven to be tighter when there are less than 35 hopping channels. Finally, we provide a concise performance comparison between LoRa-CSS and LR-FHSS in DtS-IoT, showing that LR-FHSS significantly increases the number of devices that can simultaneously transmit during a satellite pass, thereby enhancing uplink capacity.