## 3 System Model
### Network Model  
As the IoT LoRa network shown in Figure 2, we considered a classic uplink LoRa system with a single GW at the centre of a network field of radius \(R=6\) km and a set of a lot of LoRa devices (LDs), which was defined by \(LD=n_{i}|I\in\{1,\ldots,N\}\), \(N=|LD|\). LoRa devices were allocated in Gaussian distribution within the network field. For the proposed SFPCR scheme, we divided the IoT LoRa network into two regions: a low spreading factor zone (LSFZ) and a high spreading factor zone (HSFZ). In our network model, a LoRa device located in the HSFZ joins the CH in a cluster through a single hop communication and sends its data packet to the CH. Then, the CH forwards data packets from LoRa devices in its cluster to a selected relay LoRa device in the LSFZ, which transmits these data packets in a single-hop communication to the GW, as shown in Figure 2.  
The performance of LoRaWAN is restricted by a duty cycle in a sub-band for the amount of time spent transmitting packets, which is referred to as the time on air (ToA). As a consequence of this, we took into consideration a duty cycle (\(\delta_{i}\)) of 1% [4], operating at a certain frequency (Fc) in the 868 MHz band with a code rate (CR) of 4/5 and a variable bandwidth (BW) ranging from 125 kHz to 500 kHz. We explicitly deployed class A and B LoRa devices, with LoRa devices always initiating transmission. During a time slot, each LoRa device \(n_{i}\) with matching SF \(\in\) [7, 8, \(\ldots\), 12] and values for signal-to-noise ratio (SNR), receiver sensitivity for a specific bandwidth, and bit rate, as shown in Table 1, transmits a packet. The LoRa devices use different SFs for transmission to ensure orthogonality and multiple detections at the receiver.  
In [38], the authors alluded that different SFs caused significantly different times on air for symbol transmission. A symbol can encode \(k\) information bits into a chirp with \(SF=k\), and the bit rate \(R_{b}\), so the symbol period is evaluated by \(T_{symbol}=\frac{2^{k}}{BW}\), where BW is the bandwidth. When \(SF=k+1\), one symbol period is \(\frac{2^{n+1}}{BW}\), which doubles the transmission time by sending only one more bit. However, a higher SF indicates a greater resistance to interference and noise, resulting in a greater communication range.  
According to recent research, we assumed that an IoT LoRa network was divided into multiple annular rings of the same width. Thus, the width of the annulus was defined  
Figure 2: Model of the relay and cluster process in the proposed SFPCR scheme.  
here as the Euclidean distance between the two closest points on the inner boundary \(C_{in}\) and the outer boundary \(C_{out}\), respectively. Recent work has shown a spreading factor distribution scheme based on a certain range of distances to the GW. According to [22; 39], the spreading allocation is primarily based on the power level of the signal that the GW receives from the devices and the GW sensitivity without taking the device location into account. Consequently, it is not trivial to consider a general coverage contour metric to determine the degree of coverage at any point within the IoT LoRa network while avoiding interference from obstacles. In contrast, the proposed SFPCR scheme determines and allocates the appropriate SF based on the transmission distance and SNR of the received packets, as described in [24]. In addition, the SF allocation technique utilized in the SFPCR scheme was modified by assuming that all LoRa devices inside the same contour lines had the same SF. Due to obstructions and interference, LoRa devices may be the same distance apart but have different SNR values. Therefore, these devices cannot share an SF.