## 4 Operation Analysis  
At this point, we have to mention that we do not seek to make a new analysis for geographical forwarding systems that utilize the joint SNR and distance metric [18]. Rather, we want to elucidate the behavior of the major components of ROR. In line with this, for all channels in the chosen route that fulfill Equation (8), as specified from the application layer, the expected hop count to the sink can be approximated as [21] for the geographical forwarding operation of the protocol,  
\[E\left[n_{hop}(D)\right]\simeq\frac{D-d_{r}}{E\left[\varsigma(D)\right]}+1 \tag{13}\]  
where \(D\) is the distance from the source node to the sink node, \(d_{r}\) represents the approximated transmission range and \(E\left[\varsigma(D)\right]\) is the expected hop distance. As shown in Figure 9, we consider a node, \(n_{j}\), in the infinitesimal area \(dA=d\varphi d\psi\) with respect to the sink. The distance between \(n_{i}\) and \(n_{j}\), \(r_{(i,j)}\) can be computed thusly:  
\[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]  
Additionally, the expected hop distance, \(E\left[\varsigma(D)\right]\), can also be computed as:  
\[E\left[\varsigma(D)\right]=\int_{\psi_{\min}}^{D}\int_{-\varphi_{\psi}}^{ \varphi_{\psi}}r_{(i,j)}dP\left\{\mathbb{N}_{i}=j\right\} \tag{15}\]  
where \(dP\left\{\mathbb{N}_{i}=j\right\}\) is the probability that \(n_{j}\) is selected at the next hop \(\mathbb{N}_{i}\), \(\psi_{min}=D-d_{r}\) and \(\varphi_{\psi}\)=\(\mathrm{acos}\left[(\psi^{2}+D^{2}-d_{r}^{2})/2\psi D\right]\). Remember that, according to Equation (12), this means that is the closest node to the sink among all nodes that fulfil \(d_{sk}\leq d_{sk}^{fd}\) in the priority region that have retuned a one for the \(I_{DIFD}\) operation. Also, according to Equation (12), \(n_{j}\) must satisfy \(\xi_{j}\geq\xi_{th}\) to be selected as the next hop, which means that all nodes, \(n_{x}\), closer to the sink than \(n_{j}\) received the packet with an SNR value less than the threshold, _i.e._, \(\xi_{x}<\xi_{th}\). It is thus derived that the probability of selecting \(n_{j}\) as the next hop is:  
\[dP\left\{\mathbb{N}_{i}=j\right\}=P\left\{N_{A(d\psi)}=1\right\}P\left\{\xi_ {j}\geq\xi_{th}\right\}P\left\{r_{(j,s)}\geq\psi\right\} \tag{16}\]Figure 8: Flow chart for ROR node forwarding algorithm. CC, control channel; SIFS, short inter-frame space; DIFD, data initiative forwarding determination. \(P\left\{N_{A(d\psi)}=1\right\}\) represents the probability that there is a node inside the area, \(A(d\psi)\), where \(N_{A(d\psi)}\) is the number of nodes in the area, \(dA\), at distance \(\psi\) from the sink. As \(d\psi\to 0\), it can be approximated by:  
\[P\left\{N_{A(d\psi)}=1\right\}\simeq 1-e^{-\rho\psi d\psi d\varphi}\simeq\rho\psi d \psi d\varphi \tag{17}\]  
Since \(\left(\rho\psi d\psi d\varphi\right)\to 0\) as \(d\psi\to 0\) and \(d\varphi\to 0\), the approximation \(e^{-x}\simeq 1-x\) is used for the last hop. On the other hand, \(P\left\{\xi_{j}\geq\xi_{th}\right\}\) in Equation (17) is the probability that the received SNR of \(n_{j}\) is above \(\xi_{th}\); while \(P\left\{r_{(j,s)}\geq\psi\right\}\) is the probability that is at least a distance, \(\psi\), from the sink, \(s\). \(P\left\{\xi_{j}\geq\xi_{th}\right\}\) and \(P\left\{r_{(j,s)}\geq\psi\right\}\) can be derived with the log-normal channel model [11], which defines the power of a receiver at a distance, \(r\), from a transmitter as:  
\[P_{r}\left(r\right)=P_{t}-PL\left(r_{0}\right)-10\eta{\rm log}_{10}\left( \frac{r}{r_{0}}\right)+X_{\sigma} \tag{18}\]  
where \(P_{t}\) in dBm represents the transmit power, \(PL\left(r_{0}\right)\) in dB is the path loss at a reference distance, \(r_{0}\), \(\eta\) represents the path loss exponent, while \(X_{\sigma}\) represents the shadow fading component, with \(X_{\sigma}\)\(\sim\)\(\aleph\left(0,\sigma\right)\). At the receiver, the signal-to-noise ratio (SNR) in dB is given by \(\xi\left(r\right)=P_{r}\left(r\right)-P_{n}\). where \(P_{n}\) is the noise power in dBm.