## 4 Operation Analysis
Thus, taking into consideration the shadow fading component, \(X_{\sigma}\), \(P\left\{\xi_{j}\geq\xi_{th}\right\}\) becomes:  
\[P\left\{\xi_{j}\geq\xi_{th}\right\}=P\left\{X_{\sigma}\geq\beta\left(r_{(i,j) },\xi_{th}\right)\right\}\]  
\[=Q\left(\frac{\beta\left(r_{(i,j)},\xi_{th}\right)}{\sigma}\right) \tag{19}\]  
\[with\beta\left(r_{(i,j)},\xi_{th}\right)=\xi_{th}+P_{n}-PL\left(r_{0}\right)+ 10\eta{\rm log}_{10}\left(\frac{r_{(i,j)}}{r_{0}}\right) \tag{20}\]  
and \(Q\left(x\right)=\frac{1}{\sqrt{2\pi\left(\int_{x}^{\infty}e^{-\left(x^{2}/2 \right)}\right)}}dt\). Then, by representing the area wherein the nodes that are closer to the sink node than as \(A\left(\psi\right),P\left\{r_{(j,s)}\geq\psi\right\}\) can subsequently be derived as:  
\[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]0  
\[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]1 Reference model used for derivations. \[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]2 \[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]3  
in this respect, \(A(\psi)\) represents the area of intersection of two circles, which cuts across two neighboring virtual clusters, with centers separated by \(D\) and with radii \(d_{r}\) and \(\psi\), respectively. Thus, \(N_{A(\psi)}\) represents the number of nodes in the area, \(A\left(\psi\right)\), while \(M\left(\psi\right)\) represents the average number of nodes in the area, which is \(=\rho A\left(\psi\right)\). \(p_{x}\left(\psi\right)\), which represents the probability that the received SNR, \(\xi_{x}\), for \(n_{x}\) in \(A\left(\psi\right)\), is less than the SNR threshold, \(\xi_{th}\). \[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]4  
where \(\psi_{\min}=D-d_{r}\). Hence, the expected hop distance, \(E\left[\varsigma(D)\right]\), can be derived by using Equations (16), (17), (19), (21) and (22) in Equation (15) to give:  
\[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]5  
Alongside this, a simulation set-up of a single stream flow from one source node to the sink allows a clearer view of the nature of the protocol. As observed in Figure 10, the expected distances between the LMR nodes along the selected routes gradually decrease as the number of nodes in the network increases. This is because the route request phase, as controlled by Equation (11), is basically a carefully controlled flooding operation to find all possible routes to the sink via CC. This strategy is typical to most previously proposed protocols, like SCEEM and ECR. With such schemes, it is not possible to stipulate the SNR reception limit criteria for packet forwarding. This is because both next hop and channel negotiation are already determined in the route request phase via CC, while the real packet forwarding process is done via a different data channel. However, via disconnecting the channel selection from the  
\[r_{(i,j)}=r\left(D,\psi,\varphi\right)=\sqrt{\psi^{2}+D^{2}-2\psi D\cos\varphi} \tag{14}\]6 Expected hop distance _vs_. the number of CRSN nodes in the network. next hop selection in the route request and by the formation of VCGs along the path to source in the route select phase, the SNR reception limit criteria for packet forwarding can be used in order to maximize the opportunity of extending the next hop distance, since nodes in the transitional region usually exhibit the highest forwarding values for lossy links [18]. Thus, as observed from the Figure 10, the forwarding phase has the potential of selecting nodes that make more progress in the expected hop distance towards the sink, which, in summary, agrees with theoretical deduction. The value for \(\xi_{th}\) used was 10 dB, while the values in Table 1 used for analytical purposes are gotten from Mica2 [40], which corresponds with our simulation environment, Prowler [34, 41].