## 4 Multi-Hop Data Forwarding Strategy Based on Annulus-Sectors
### The Length of Radius of the RCCH  
As mentioned above, the load on the CHs that is close to the center of the network is relatively higher although the area of the annulus-sector in which it is located is relatively smaller. It needs to forward a large amount of data uploaded by the CHs in the outer annuli. Therefore, for annulus-sectors in different annuli, the sizes of their RCCH should be different to balance the energy consumption of the whole network. In order to ensure the lifetime of each RCCH is approximately equal to each other, the following constraint needs to be met:  
\[\pi\tau_{1}^{2}\rho e_{0}/E_{CH}^{1}\approx\pi\tau_{2}^{2}\rho e_{0}/E_{CH}^{2 }\approx\ldots\approx\pi r_{t}^{2}\rho e_{0}/E_{CH}^{i}\approx\pi\tau_{R}^{2} \rho e_{0}/E_{CH}^{k} \tag{13}\]  
In (13), \(\pi\tau_{i}^{2}\rho e_{0}\) is the total initial energy of the nodes in one RCCH of the \(i\)-th annulus. After simplification, that is:  
\[r_{i}/r_{i+1}\approx\sqrt{E_{CH}^{i}/E_{CH}^{i+1}} \tag{14}\]  
When \(i\in[1,k-2]\), according to Equations (10) and (14), it can be obtained that:  
\[\frac{r_{i}}{r_{i+1}}\approx\sqrt{1+\Bigg{(}(N_{i}e_{r}+N_{i}e_{t})\Big{/} \left(\Bigg{(}\sum\limits_{m=i+1}^{k}N_{m}-1\Bigg{)}e_{r}+\sum\limits_{m=i+1}^ {k}N_{m}e_{t}\Bigg{)}\Bigg{)}} \tag{15}\]  
Since \(N_{i}\) is always less than \(N_{i}+1\), the ratio between \(r_{i}\) and \(r_{i}+1\) is larger than \(1\) but smaller than \(2\). That is, the closer the RCCH is to the network center, the larger its area is. When \(i=k-1\), after substituting Equation (10) into Equation (14), it can be obtained that:  
\[\frac{r_{k}}{r_{k-1}}\approx\sqrt{\frac{(N_{k}-1)e_{r}+N_{k}e_{t}}{(N_{k-1}+N _{k}-1)e_{r}+(N_{k-1}+N_{k})e_{t}}} \tag{16}\]According to Equation (11), it is known that \(N_{k-1}=(2k-3)/(2k-1)\)\(N_{k}\). By combining this equation and Equation (16), we can get:  
\[\frac{r_{k}}{r_{k-1}}\approx\sqrt{\left((N_{k}-1)e_{r}+N_{k}e_{t}\right)\Big{/} \left(\left(\left(\frac{4k-4}{2k-1}\right)N_{k}-1\right)e_{r}+\left(\frac{4k-4 }{2k-1}\right)N_{k}e_{t}\right)} \tag{17}\]  
It is not difficult to know that the ratio between \(r_{k}\) and \(r_{k-1}\) is smaller than 1, so the area of the RCCH in the outermost annulus-sectors is the smallest as the CHs in these annulus-sectors need not forward data from other annulus-sectors. In this case, the burdens on these CHs are relatively light so there is no need for too many nodes to participate in CH selection. According to Equation (13), the length of radius of the RCCH in the \((k-1)\)-th annulus is important for balance of energy consumption between CHs, because it affects not only the value of \(r_{k}\), but also the value of \(r_{i}\) (\(i\in[1,k-2]\)). For the annulus-sector in the \(k\)-th annulus, there must be at least one node in its RCCH. When \(\pi r_{k}^{2}\rho>1\), the above constraint can be satisfied. That is to say, the following formula must be met:  
\[r_{k}\geq 1/\sqrt{\pi\rho} \tag{18}\]  
On the other hand, the range of the RCCH should not go beyond the boundary of the annulus-sector at which it is located. There is no doubt that the area of the annulus-sector in the first annulus is the smallest, and according to Equations (15) and (17), it is also known that the area of the RCCH in this annulus is the largest. Thus, we only care about the upper limit of \(r_{1}\).