Unified Spatial Analytical Framework for Confirmed Uplinks in Multi-Gateway LoRaWAN Under Duty-Cycle Constraints

Confirmed uplinks in LoRaWAN improve reliability by requiring acknowledgment feedback from the network. However, gateway constraints such as half-duplex operation and duty-cycle limitations often prevent timely acknowledgment delivery, triggering retransmissions and degrading performance. These challenges are compounded in multi-gateway deployments, where overlapping coverage, correlated reception, and spatial interference create tightly coupled uplink-downlink dynamics. While prior analytical models capture parts of this behavior, a unified framework that jointly accounts for multi-gateway topology, heterogeneous bi-directional traffic, and spatial interference remains limited. In this paper, we propose a Unified Spatial Analytical Framework for confirmed-uplinks in multi-gateway LoRaWAN under percentage-based duty-cycle constraints. The framework serves as a fast and scalable what-if diagnostic engine for network planning and configuration. It combines a hex-cell spatial abstraction with imperfect spreading-factor orthogonality and heterogeneous traffic classes (NAPP/PAPP/CAPP), and couples these with a bi-directional M/G/1/1 renewal model to capture gateway transmission availability. Validation against NS-3 shows high fidelity in predicting key performance indicators (uplink success, downlink success, and retransmission overhead), with errors below 10% at fine spatial resolution. Finally, diagnostic case studies reveal that performance collapse is primarily driven by downlink capacity starvation (rather than uplink collisions) and expose the limitation regimes and fairness trade-offs of representative mitigation levers, including load-balance downlink assignment, gateway densification, and downlink airtime reduction.