# Dynamic Cooperative MAC Protocol for Navigation Carrier Ad Hoc Networks: A DiffServ-Based Approach
## 5 Performance Analysis
### Searching Probability of Next-Hop Nodes  
In OC mode, the throughput has closed correlations with the location of next-hop nodes [21]. Thus, we should analyze the probability of searching next-hop nodes first. This probability is deduced in single-hop (Scenarios 1 and 2) and 2-hop (Scenario 3), respectively. **Lemma 2**. : _Given node \(C_{i}^{(j)}\) (\(j=0,1\)), the next-hop relay of \(C_{i}^{(j)}\) should satisfy two constraints: (i) node \(C_{i}^{(j+1)}\) locates in three-dimensional sensing region of \(C_{i}^{(j)}\) when the data rate is \(x\in\{1,2,5.5,11\}\) Mbps, (ii) the PDR is affected by fading effects and interference._  
In Scenarios 1 and 2, node \(C_{i}^{(j+1)}\) is within the transmission range of node \(C_{i}^{(j)}\) when the data rate is \(x\) Mbps; thus, node \(C_{i}^{(j)}\) finds it easy to find node \(C_{i}^{(j+1)}\) in DT-OR mode (Scenario 1) or OT-OR mode (Scenario 2). The node \(C_{i}^{(j+1)}\) will decode the azimuth and localization of \(C_{i}^{(j)}\) from the RTS packet, and reply CTS to \(C_{i}^{(j)}\) with a delivery ratio of approximately 100%, thus the next-hop searching probability in Scenarios 1 and 2 can be derived as:  
\[P_{1-hop}(x,y,l)=\left\{ \begin{array}{cl}1-\beta,&l\in[l_{\textit{threshold}}, \overline{d}),\ \ \text{Scenario 1}\\ (1-\beta)V_{dir}(d_{x},\theta_{\max})/V_{\textit{omni}}(d_{x}),&l\in[l_{ \textit{threshold}},\overline{d}),\ \ \text{Scenario 2}\end{array} \right. \tag{6}\]  
Figure 6: Potential problematic scenarios in DDC-MAC. (**a**) problematic scenario of deafness and hidden terminal problem; (**b**) solutions of deafness and hidden terminal problem by DDC-MAC; (**c**) problematic scenario of interference and congestion problem; (**d**) solutions of interference and congestion problem by DDC-MAC. where \(l_{threshold}\) is the minimum unperturbed distance between \(C_{i}^{(j)}\) and \(C_{i}^{(j+1)}\), \(V_{dir}(d_{x},\theta_{\max})\) is the coverage region of directional beam of \(C_{i}^{(j)}\) when its beamwidth is \(\theta_{\max}\) and transmission range is \(d_{x}\), \(\beta\) is the fading effects and interference on PDR, \(V_{omni}(x)\) is the detection region of directional beams (with data rate \(x\)), which can be expressed as follows:  
\[V_{dir}(d_{x},\theta_{\max})=\frac{2}{3}\pi d_{x}^{3}\bigg{[}1-\cos\frac{ \theta_{\max}}{2}\bigg{]},\,V_{omni}(d_{x})=\frac{4}{3}\pi d_{x}^{3} \tag{7}\]  
In Scenario 3, receiver \(C_{i}^{(j+2)}\) exceeds the direct range of its sender \(C_{i}^{(j)}\), so only the node \(C_{i}^{(j+1)}\), which is located in the ranges of both \(C_{i}^{(j)}\) (with data rate \(x\)) and \(C_{i}^{(j+2)}\) (with data rate \(y\)), can be used as a next-hop node. In other words, \(C_{i}^{(j+1)}\) should be located in the overlapping region of \(C_{i}^{(j)}\) and \(C_{i}^{(j+2)}\) when the distance is \(l\) km (\(l\leq d_{x}+d_{y}\)). The transmission mode of Scenario 3 includes DT-DR-OR mode and OT-OR-OR mode, as shown in Figure 7, thus only the node \(C_{i}^{(j+1)}\) within the hatch area can meet the conditions in Lemma 2. Suppose that the distance between \(C_{i}^{(j)}\) (with data rate \(x\)) and \(C_{i}^{(j+2)}\) (with data rate \(y\)) is \(l\) km (\(l\leq d_{x}+d_{y}\)), the volume of the hatch area \(V_{0}(d_{x},d_{y},l)\) in Figure 7 can be calculated as  
\[V_{0}(d_{x},d_{y},l)=\frac{\pi}{6}d_{x}^{3}\cdot\Big{[}4-3\sin^{3}\frac{\theta _{1}}{2}-\cos^{3}\frac{\theta_{1}}{2}+3\cos\frac{\theta_{1}}{2}\Big{]}+\frac{ \pi}{6}d_{y}^{3}\cdot\Big{[}4-3\sin^{3}\frac{\theta_{1}}{2}-\cos^{3}\frac{ \theta_{2}}{2}+3\cos\frac{\theta_{2}}{2}\Big{]} \tag{8}\]  
where \(\theta_{1}\) is beamwidth angle of node \(C_{i}^{(j)}\), \(\theta_{2}\) is beamwidth angle of \(C_{m}^{(j+2)}\). In DT-DR-OR mode, \(\theta_{1}\in[0,\theta_{\max}]\) and \(\theta_{2}\in[0,\theta_{\max}]\), whereas in OT-OR-OR mode, \(\theta_{1}\in[0,\pi)\) and \(\theta_{2}\in[0,\pi)\).