## 3 Related Work
The work enhanced the achieved throughput and had a smaller number of packet transmissions than a simple peer-to-peer, multi-hop topology with specific delay tradeoffs. The work in [36] focused on the problem of delay and throughput in linear LoRa multi-hop networks. The results showed an improvement in throughput, but when the number of sensors increased, the network performance was affected, thus impacting the system scalability and the sensor energy. In [37], the authors proposed an underground monitoring system. They considered a linear multi-hop network; they also studied an energy model for evaluating the energy cost per sensor and presented a synchronization method to enhance the duty cycle and reduce energy consumption without considering the network throughput. We use the same sensor types. The following works performed an analytical evaluation of the network performance. Throughput can be improved by editing the network configurations (parameters) such as packet size, transmission range, and the number and distribution of sensors. Selecting the best network parameters (such as payload size) to provide high throughput provides good results, as in [38], where the authors introduced a mathematical model to assess the achieved throughput of NB-IoT technology. The work in [38] focused on network planning with the aid of an analytical model that estimated the network capacity throughput and network coverage using different network scenarios (different network parameters such as payload size). In our work, we follow the same methodology for the LoRaWAN analytical model to estimate the capacity. There was no energy consumption assessment. In a similar work [39], the model assessed the network coverage, throughput, and scalability using different scenarios and experimental verification. There was no energy assessment. In [40], the authors investigated different scenarios (various numbers of sensors and different distributions) to provide optimal throughput of LoRaWAN technology. All scenarios were single-hop; we derived our single-hop scenario from this study. Additionally, a mathematical model for assessing the performance of a single-hop LoRaWAN was introduced. The works in [10; 41] studied the interdependence between application requirements and network configurations because of the diversity of data traffic, environmental conditions, and coverage range (distance between sensors), showing that uneven sensor density and distribution limit the scalability of single-hop LoRaWAN. Both studies involved an analytical model, the model in [10] studied a multi-hop LoRaWAN in general, and in [41] they evaluated the network capacity and throughput for NB-IoT. All the analytical studies were conducted on single-hop LoRaWAN except [10]. Since our work involves measuring performance in a LoRaWAN multi-hop network, the only research that is analytical and studies the multi-hop is the one in the [10]. This work also studies similar parameters to our research, i.e., energy consumption, network capacity, and throughput. However, we consider the pipeline requirements in the model, and the sensor in our model plays the role of a replay device dissimilar to the work in [10]. However, their results show an energy issue in high-traffic scenarios. To evaluate performance, the studies used three different methods: real experiments, simulation, and analytical evaluation (mathematical approach). To conclude, some of the studies presented earlier addressed the LoRaWAN star or hierarchical networks. Furthermore, many multi-hop LoRaWAN studies did not consider the application requirements and network throughput. Also, energy consumption was not widely studied. However, most of the related work used experiments or simulations. In this work, we propose a multi-hop linear LoRaWAN model that evaluates the performance of a long linear sensor topology. The studied system has a high sensor density, may involve high network traffic, and considers different network architectures (scenarios).