# Optimized Dynamic Deployment of UAVs in Maritime Networks with Route Prediction
## 3 System Modeling
### Network Communication Model  
At the \(t\)-th time slot, the position of the \(m\)-th UAV can be expressed as:  
\[w_{m}(t)=(x_{m}(t),y_{m}(t),h) \tag{1}\]  
\[x_{m}(t+1)=x_{m}(t)+\cos\theta\cos\phi d_{m}(t) \tag{2}\]  
\[y_{m}(t+1)=y_{m}(t)+\sin\theta\cos\phi d_{m}(t) \tag{3}\]  
where \(x_{m}(t)\) denotes as the horizontal coordinate of the \(m\)-th UAV at t time slots and \(y_{m}(t)\) denotes the vertical coordinate of the \(m\)-th UAV at the time of t time slots, and \(h\) denotes the height of the \(m\)-th UAV from the sea surface at the time of t time slots. In this paper, the altitude of the flight varies between \(h_{1}\) and \(h_{2}\). \(x_{m}(t+1)\) denotes the horizontal coordinate of the \(m\)-th UAV at \(t+1\) time slots, and \(y_{m}(t+1)\) denotes the vertical coordinate of the \(m\)-th UAV at \(t+1\) time slots. \(d_{m}\in[d_{\min},d_{\max}]\) denotes the distance flown by the drone at a one-time slot. \(d_{\max}\) denotes the maximum flight distance of the drone in a one-time slot. \(\theta\in[0,2\pi]\) denotes the horizontal flight direction. The linear distance between UAV m and user n is as follows, where \(x_{n}(t)\) denotes the horizontal coordinate of user n at t time slots, and \(y_{m}(t)\) denotes the vertical coordinate of user n at t time slots:  
\[L_{(m,n)}(t)=\sqrt{h_{m}(t)^{2}+(x_{m}(t)-x_{n}(t))^{2}+(y_{m}(t)-y_{n}(t))^{2}} \tag{4}\]  
In our model, it is assumed that the satellite and terrestrial base station TBS are relatively stationary within the shorter time slot \(T_{\text{set}}\). At the moment \(t\), the position of the satellite is denoted as \((x_{\text{[PERSON]}}(t),y_{\text{[PERSON]}}(t),h_{\text{[PERSON]}}(t))\), and the position of the ground base station is denoted as \((x_{\text{tbs}}(t),y_{\text{tbs}}(t))\). The straight-line distance between the satellite, ground base station, and user n is denoted as:  
\[L_{\text{[PERSON],}n}(t)=\sqrt{h_{\text{[PERSON]}}(t)^{2}+(x_{\text{[PERSON]}}(t)-x_{n}(t))^{ 2}+(y_{\text{[PERSON]}}(t)-y_{n}(t))^{2}} \tag{5}\]  
\[L_{\text{tbs,}n}(t)=\sqrt{(x_{\text{tbs}}(t)-x_{n}(t))^{2}+(y_{\text{tbs}}(t)- y_{n}(t))^{2}} \tag{6}\]  
Figure 1: Maritime communication network model. Assuming that there is no mission migration between UAVs, satellites, and base stations, and that the minimum channel capacity \(C\) and bandwidth \(B\) of each device are known, we can find the minimum signal-to-interference-plus-noise-ratio (SNR) according to [PERSON]'s formula:  
\[\zeta_{\text{UAV,min}}(t) =2\frac{\zeta_{\text{UAV,min}}(t)}{\zeta_{\text{UAV}}(t)}-1 \tag{7}\] \[\zeta_{\text{leo,min}}(t) =2\frac{\zeta_{\text{leo,min}}(t)}{\zeta_{\text{hoo}}(t)}-1\] (8) \[\zeta_{\text{tbs,min}}(t) =2\frac{\zeta_{\text{tbs,min}}(t)}{\zeta_{\text{tbs}}(t)}-1 \tag{9}\]  
Assume that all UAVs share the same frequency band, so the UAVs interfere with each other in downlink transmissions [36]. Meanwhile, it is assumed that the transmit power \(P_{t}\) from all UAVs to the user and the channel gain \(G_{0}=\frac{(G_{t}G_{r})}{\zeta_{\text{UAV,max}}}\) are the same, so the signal-to-noise ratio \(\varsigma\) can also be expressed as the following equation, where \(n^{2}\) denotes the noise power at the receiving end:  
\[x_{m}(t+1)=x_{m}(t)+\cos\theta\cos\phi d_{m}(t) \tag{2}\]0  
Therefore, the maximum coverage radius of the UAV can be derived as:  
\[x_{m}(t+1)=x_{m}(t)+\cos\theta\cos\phi d_{m}(t) \tag{2}\]1  
Since this system assumes that there is only one satellite and base station and they are in separate frequency bands, the interactions between the UAV, satellite, and base station are negligible when calculating the signal-to-noise ratio.