Coverage Probability Analysis and Relay Placement Optimization for Two-Hop LoRa Networks with Random Traffic Activation

In LoRa uplink communication, direct edge-node-to-gateway transmission is affected by path loss, thermal noise, small-scale fading, and intra-spreading-factor (intra-SF) and inter-spreading-factor (inter-SF) interference under random traffic activation. These factors reduce the signal-to-interference-plus-noise ratio (SINR) and degrade coverage reliability. To address these issues, this study proposes an integrated framework for coverage probability analysis and relay placement optimization in two-hop LoRa networks with random traffic activation. First, a two-hop LoRa uplink network model is established, consisting of edge source nodes, a decode-and-forward (DF) relay, a central gateway, and potential interfering nodes. By incorporating distance-dependent path loss, receiver-side thermal noise power, and small-scale fading gains, a unified received-power model is formulated for the desired and interfering links. Second, a Bernoulli traffic-activation indicator is assigned to each potential interfering node to characterize its random transmission state and link the traffic activation probability, the active-interferer set, and the expected number of active interferers. An interference model is then developed to quantify the effect of random traffic activation on aggregate interference over communication links. To account for multi-spreading-factor (multi-SF) coexistence in LoRa, intra-SF interference and residual inter-SF interference are incorporated into the link-level SINR criterion, and the corresponding single-hop coverage probability is derived. Finally, based on the DF relaying protocol, successful end-to-end transmission is modeled as the joint event of successful first-hop decoding and successful second-hop forwarding, and a two-hop coverage probability model is constructed. A system-level coverage probability model is also developed to capture the complementarity between direct transmission and two-hop relaying. A relay placement optimization problem is then formulated to maximize the weighted average system coverage probability across multiple traffic states. The performance of fixed, random, and optimized relay placement schemes is compared. Simulation results demonstrate that the proposed random-traffic-aware relay placement optimization method substantially improves the end-to-end coverage probability compared with fixed and random relay placement schemes, thereby enhancing the communication reliability of edge-node transmissions.