# A Cross-Layer Optimized Opportunistic Routing Scheme for Loss-and-Delay Sensitive WSNs
## 5 The Design of the COOR Scheme  
### Transmission Power Optimization  
The COOR scheme is the improvement of the basic network model. Thus, it is necessary to calculate the data volume and the energy consumption of each tier in the wireless sensor network under constant power. Then, according to this information, the node transmission power of different areas is adjusted to achieve the purpose of optimization. The calculation of data volume is to count the number of data packets that each node should bear. The exact number of packets is critical in computing the energy consumption. Therefore, the following describes how to calculate the number of packets that each node bears. The basic network transmission structure is given in Figure 9. The radius of the circular network is \(R\), and the node transmission radius is \(r\). The nodes in the network are evenly distributed and the density is \(\rho\). The area \(A_{l,k}\) is a fan-shaped ring area with a distance \(l\) to the sink, a width \(d_{x}\), and a radians value \(d_{\theta}\). The area \(A_{l+x,k+j}\) is \(x\) far from \(A_{l,k}\) with the difference in the clockwise radians \(j\)\(\cdot\)\(d_{\theta}\). **Theorem 1**. : _Consider a small fan-shaped ring region \(A_{l,k}\), the total number of nodes contained in this area is as:_  
\[S_{l,k}=pld_{\theta}d_{x} \tag{18}\]  
**Proof. ** As shown in Figure 9, the area of region \(A_{l,k}\) can be calculated by the following formula:  
\[s=\frac{1}{2}(l+d_{x})^{2}d_{\theta}-\frac{1}{2}l^{2}d_{\theta}\]  
The number of nodes is equal to the distribution density multiplied by the area:  
\[S_{l,k}=\rho s=\rho ld_{\theta}d_{x}\]  
\(\Box\)  
Further analysis of the data volume of a specific node in different regions is as follows: consider the edge area of the network (\(R-r<l\leq R\)), the nodes in this part of the network do not undertake the forwarding of packets from any other node, so the number of packets sent is 1, the number of packets received and the number of ACK packets sent are 0. At the same time, the number of CTS packets it sends is 1, which is used to confirm the communication clear before the next hop node replies ACK packet. For non-edge areas in the network, as shown in Figure 10, region \(A_{l,k}\) will rely all packets sent from \(A_{l+r,k}\). If the partition is small enough, then for any node \(n_{l+r}\) in the region \(A_{l+r,k}\), it is the same to choose any node \(n_{l}\) in \(A_{l,k}\) in the aspect of transmission reliability. Figure 10: Calculation diagram of data volume. Figure 9: Network transmission structure. **Theorem 2**. : _When the success rate of a single transmission between node \(n_{l+r}\) and node \(n_{l}\) is \(p_{l+r}\), in a round of sending each node sends a packet to the sink. Then, respectively, the number of sending packets \(\mathcal{Q}^{S}(l)\), received packets \(\mathcal{Q}^{R}(l)\), ACK packets \(\mathcal{Q}^{A}(l)\) and CTS packets \(\mathcal{Q}^{C}(l)\) of node \(n_{l}\) can be calculated as:_  
\[\mathcal{Q}^{S}(l)=\ \frac{l+r}{l}\cdot(1-(1-p_{l+r})^{N})\cdot\mathcal{Q}^{S}(l+r )+1 \tag{19}\]  
\[\mathcal{Q}^{R}(l)=\ \frac{l+r}{l}\cdot N\cdot\mathcal{Q}^{S}(l+r) \tag{20}\]  
\[\mathcal{Q}^{C}(l)=\mathcal{Q}^{S}(l)+\mathcal{Q}^{R}(l)(1+\frac{1}{p_{l+r}}- \frac{1-(1-p_{l+r})^{N}}{Np_{l+r}}) \tag{21}\]  
\[\mathcal{Q}^{A}(l)=\mathcal{Q}^{R}(l)\cdot\frac{1-(1-p_{l+r})^{N}}{Np_{l+r}} \tag{22}\]  
Proof. : All the nodes under the traditional routing method have the same transmission radius, so the transmission between the areas is periodic. As shown in Figure 9, region \(A_{l,k}\) assumes the forwarding of the data generated by the nodes in region \(A_{l+r,k},A_{l+2r,k},\cdots A_{l+mr,k}\), where \(m=\left\lfloor\frac{R-l}{r}\right\rfloor\). For the area \(A_{l,k}\), all the nodes in this area directly and uniformly assume the forwarding of data packets sent by the nodes in the outer area \(A_{l+r,k}\) with distance \(r\).