## 3 System Model
### Success Probability Model  
In this section, we examine the success probability of a signal in LoRa's respective spreading factor regions under intra- and inter-SF interference. In this paper, we investigated the average success probability of LoRa devices in random locations in the field of an IoT LoRa network in order to determine the optimal number and positions of relay LoRa devices (LDs) in relation to the distance between a source LD and the GW. The proposed SFPCR scheme considers the evaluation of a source LD's packet success probability using a varying SF to be dependent on the distance from the receiving LoRa device (i.e., the source LD (S) to the CH, the CH to the relay LD (R), and R to the GW). The collision of packets in a LoRa network is classified into two types: intra-SF collision and inter-SF collision [44]. If a transmission collides, the LoRa device becomes backlogged and slows down, resulting in excessive delay and packet loss. As a result, it is critical for the LoRa link to have a high data reliability.  
Transmission from LDs located within a distance \(\leq r\) to the GW is considered successful in a LoRa network with a uniform distribution of N nodes over the disk of density \(\lambda=\frac{N}{\pi^{2}}\) if and only if a source LD (S) transmits a packet at distance x from the GW that is not affected by either intra-SF or inter-SF collisions. Furthermore, we assumed that LoRa devices released packets within predetermined time slots within a deployment radius of r, and that packet transmission intensity was \(\theta=\frac{pkt}{\pi}\) (bits per second), where \(pkt\) is the number of packets and \(\tau\) is the time interval in seconds.  
Intra-SF interference occurs when LDs in the same spreading factor region and BW collide when the distance from the GW is less than \(xR\). Here, \(x\) is a location of a LoRA node and \(R=e^{\frac{6}{16\gamma}}>1\)[45], and the density of the LDs within the same spreading factor region \(SF(i)\) is represented by \(\rho_{i}=\frac{\alpha_{i}N}{\pi^{2}}\)[44]. Subsequently, the number of potential interferers \(I_{SF(i)}\) within the same spreading factor region (intra-SF) is represented by Equation (12).  
\[I_{SF(i)}=\alpha_{i}N\frac{(min(xR,r))^{2}}{r^{2}} \tag{12}\]  
Here, \(N\) is the number of LoRa devices in the disk, \(\alpha_{i}\) is the percentage of LDs in a particular SF with \(\sum_{\gamma}^{12}\alpha_{i}=1\). Assuming that none of the potential interfering LDs initiate a transmission during the vulnerable period of \(2T_{i}\) duration, \(P_{suc}^{i}(loc(x))\) is the probability of successful transmission of a LoRa device at location \(x\) in an IoT network area and is represented by the Equation (13),  
\[P_{suc}^{i}(loc(x)):=e^{-2T_{SF(i)}\theta I_{SF(i)}} \tag{13}\]  
where \(T_{SF(i)}\) is the time on air evaluated in Equation (2).  
Consequently, the combined success probability \(P_{suc}(S,GW)\) from the source LD (S) to the GW in the proposed SFPCR scheme takes advantage of the varying SF as defined in (14):  
\[P_{suc}(S,GW)=P_{suc}^{SF(i)}(S,CH)\times P_{suc}^{SF(i)}(CH,R)\times P_{suc}^{ SF(k)}(R,GW) \tag{14}\]  
Here, \(S\) represents the source LoRa device \(n_{i}\), CH is the cluster head represented as \(CH_{i}\), R is the relay LoRa device also represented as \(N_{i}^{*}\), and GW is the gateway, respectively. Then, \(P_{suc}^{SF(i)}(S,CH)\) is the success probability from S to CH, \(P_{suc}^{SF(i)}(CH,R)\) from CH to R, and \(P_{suc}^{SF(k)}(R,GW)\) from R to GW. \(SF(i)\), \(SF(j)\), and \(SF(k)\) are the spreading factors assigned to each of them.