# Research on Communication Stability of Inter-Cannonball Network Based on OPNET
## 2 Theoretical Modelling
### Modelling of Analytical Methods  
In order to analyse the route establishment process, real-time data transmission and network overhead of the communication network are established. We collect and analyse the number of hops per route, route discovery time, wireless LAN delay, and throughput parameters from the modelling simulation data [25; 26].  
Throughout is usually expressed as the amount of data sent and received by a node per unit of time and is defined as:  
\[X=\frac{C}{\tau} \tag{6}\]  
where \(C\) is the total number of tasks completed, and \(\tau\) is the total time taken to complete these tasks. For statistical convenience, the number of packets that successfully reach the destination node and are received at the upper layer is used as the throughput data.  
The inter-cannonball ad hoc network is a more delay-sensitive network, where the delay \(\overline{D}_{\mathrm{E2E}}\) is the sum of the delays of all links on the established route, defined as follows:  
\[\overline{D}_{\mathrm{E2E}}=\frac{\sum D_{i,\mathrm{E2E}}}{N_{\mathrm{R}}}= \frac{\sum(t_{i,\mathrm{R}}-t_{i,\mathrm{S}})}{N_{\mathrm{R}}} \tag{7}\]  
where \(t_{i,\mathrm{S}}\) and \(t_{i,\mathrm{R}}\) are the time when messages are sent from the source node and received at the destination node, respectively, and \(N_{\mathrm{R}}\) denotes the total number of messages received by  
Figure 3: Geocentric coordinate system.  
the destination node. Considered the most dominant transmission delay, let the propagation delay from the source node to the destination node be \(D_{SD}^{\text{Pro}}\). Then, it is expressed as:  
\[D_{SD}^{\text{Pro}}=\frac{L_{SD}}{c} \tag{8}\]  
where \(L_{SD}\) denotes the path length from \(node\_S\) to \(node\_D\), and \(c\) denotes the propagation speed of the signal over the channel. Using the geocentric coordinate system, we know that at any moment, the link length between two neighbouring nodes \(V_{i}\) and \(V_{j}\) is:  
\[\text{dis}\big{(}V_{i},V_{j}\big{)}=\sqrt{\left(X_{i}^{t}-X_{j}^{t}\right)^{2}+ \left(Y_{i}^{t}-Y_{j}^{t}\right)^{2}+\left(Z_{i}^{t}-Z_{j}^{t}\right)^{2}} \tag{9}\]  
where \((X_{i}^{t},Y_{i}^{t},Z_{i}^{t})\) and \((X_{j}^{t},Yj,Z_{j}^{t})\) are the coordinate positions of \(node\_i\) and \(node\_j\) at moment t, respectively. The route between \(node\_S\) and \(node\_D\) can be represented by the set of links \(E_{SD}=\big{\{}(V_{S},V_{i}),(V_{i},V_{n}),\ldots,(V_{j}V_{D})\big{\}}\), then the length of the route is:  
\[L_{\text{SD}}=\text{dis}(V_{S},V_{i})+\text{dis}(V_{i},V_{n})+\cdots+\text{dis }\big{(}V_{j},V_{\text{D}}) \tag{10}\]  
Combining Equations (9) and (10) yields an expression for the path length from \(node\_S\) to \(node\_D\) as:  
\[\begin{array}{ll}L_{\text{SD}}=&\sqrt{\left(X_{S}^{t}-X_{i}^{t}\right)^{2}+ \left(Y_{S}^{t}-Y_{i}^{t}\right)^{2}+\left(Z_{S}^{t}-Z_{i}^{t}\right)^{2}}\\ &+\sqrt{\left(X_{i}^{t}-X_{n}^{t}\right)^{2}+\left(Y_{i}^{t}-Y_{n}^{t}\right)^ {2}+\left(Z_{i}^{t}-Z_{n}^{t}\right)^{2}}\\ &+\cdots\ldots\\ &+\sqrt{\left(X_{j}^{t}-X_{\text{D}}^{t}\right)^{2}+\left(Y_{j}^{t}-Y_{\text{ D}}^{t}\right)^{2}+\left(Z_{j}^{t}-Z_{\text{D}}^{t}\right)^{2}}\end{array} \tag{11}\]