# A Cross-Layer Optimized Opportunistic Routing Scheme for Loss-and-Delay Sensitive WSNs
## 6 Performance Analysis and Comparison
### Network Weighted Average Performance  
Before presenting the results of the experiment, this section firstly introduces the calculation method of network partition reliability and delay. The reception rate of data packets and the hop count are respectively used as the measure of reliability and delay. In addition, partitions are mainly based on the distance to the sink as well as the physical location of the nodes in the network. As shown in Figure 9, in the calculation of partition performance, the circular network area is divided into multiple rings with equal width, and each ring is regard as a partition. In general, this paper uses \(pat_{x}^{y}\) to represent a partition whose distance to the sink is between \(x\) m and \(y\) m.  
**Definition 3**.: _When measuring partition performance, the weight of each node is equal. Consider that the performance of node \(n_{l}\) whose distance to sink is \(Fet^{l}\), the overall performance \(Fet_{x}^{y}\) of partition \(pat_{x}^{y}\) is calculated as:_  
\[Fet_{x}^{y}=\frac{\int_{x}^{y}\int_{0}^{2\pi}Fet^{l}\cdot\rho l\cdot d\!l\cdot d \theta}{\rho\pi y^{2}-\rho\pi x^{2}} \tag{37}\]  
**Theorem 6**.: _Consider that \(\psi(l,P_{t},r)\) denotes the reliability of the node whose distance to the sink is \(l\), the transmission radius is \(r\), and the transmission power is \(P_{t}\). Then, the overall reliability \(\psi_{x}^{y}\) of the partition \(pat_{x}^{y}\) calculated in a discrete form can be expressed as:_  
\[\psi_{x}^{y}=\frac{2\ \sum_{i=x}^{y}\psi(i,P_{t},r)\cdot i}{(y+x)\cdot(y-x+1)} \tag{38}\]  
**Proof.** Putting Equations (4), (5) and (35) together, the reliability of partition is obtained:  
\[\psi_{x}^{y}=\frac{\int_{x}^{y}\int_{0}^{2\pi}\psi(l,P_{t},r)\rho l\cdot d\!l \cdot d_{\theta}}{\rho\pi y^{2}-\rho\pi x^{2}}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Description** & **Value** \\ \hline \(R\) & Network radius & 1200 m \\ \(r\) & Transmission radius & 60 m \\ \(P_{t}\) & Transmission power & \(-3\) dBm \\ \(P_{r}\) & Reception power & Calculation \\ \(N\) & Number of candidate nodes & 2 \\ \(R_{D}\) & Sending rate/Receiving rate & 19.2 kbps \\ \(B_{N}\) & Noise bandwidth & 30 kHz \\ \(P_{n}\) & Background noise & \(-115\) dBm \\ \(d_{0}\) & Reference distance & 1 m \\ \(n\) & path loss exponent & 2 \\ \(L^{s}\) & Length of data packet & 128 Byte \\ \(L^{ACK}\) & Length of ACK packet & 8 Byte \\ \(L^{CTS}\) & Length of CTS packet & 8 Byte \\ \hline \end{tabular}
\end{table}
Table 2: Main notations and values adopted in simulation.  
Due to the discontinuity of \(\psi(l,P_{t},r)\) and the actual distribution of nodes, it is necessary to perform discretization when calculating. At the same time, it should be noted that during the conversion process, deviations may occur, which may result in a reliability value exceeding 1.  
\[\text{Assume}\ \forall i\ \psi(i,P_{t},r)=1\frac{2\pi\sum_{i=0}^{R}\psi(i,P_{t},r) \cdot\rho\cdot i}{\rho\pi y^{2}-\rho\pi x^{2}}\ =\ \frac{2\ \sum_{i=0}^{R}i}{y^{2}-x^{2}}\ =\ \frac{y-x+1}{y-x}>1\]  
After eliminating the deviation of conversion:  
\[\psi_{x}^{y}=\frac{2\pi\sum_{i=x}^{y}\psi(i,P_{t},r)\cdot\rho i}{\rho\pi y^{2}- \rho\pi x^{2}}\ \times\ \frac{y-x}{y-x+1}\]  
\[\psi_{x}^{y}=\frac{2\sum_{i=x}^{y}\psi(i,P_{t},r)\cdot i}{(y+x)\cdot(y-x)}\ \times\ \frac{y-x}{y-x+1}\]  
\[\psi_{x}^{y}=\frac{2\ \sum_{i=x}^{y}\psi(i,P_{t},r)\cdot i}{(y+x)\cdot(y-x+1)}\]  
\(\square\)  
Similarly, the overall reliability \(D_{x}^{y}\) of the partition \(pat_{x}^{y}\) calculated in a discrete form can be expressed as:  
\[D_{x}^{y}=\frac{2\ \sum_{i=x}^{y}D(i,P_{t},r)\cdot i}{(y+x)\cdot(y-x+1)} \tag{39}\]  
Here, \(D(l,P_{t},r)\) indicates the delay of the node whose distance to the sink is \(l\), the transmission radius is \(r\), and the transmission power is \(P_{t}\).