# Analysis of Maritime Wireless Communication Connectivity Based on CNN-BiLSTM-AM
## 2 The System Model
### Connectivity Probability  
Connectivity probability is one of the fundamental performance metrics in wireless communication systems. It is defined as the probability that the receiver successfully receives a signal with a signal-to-noise ratio (SNR) exceeding a specific threshold during signal transmission. Based on the wireless fading channel model described in Section 2.1, this paper primarily analyzes the distribution of the SNR and the connection probability under different path conditions, particularly in the case of Rayleigh fading. It is postulated that the channel gain \(|h|^{2}\sim\text{exp}(\lambda)\) at the receiver follows an exponential distribution. Therefore, its expression is given by  
\[f(|h|^{2})=\lambda e^{-\lambda|h|^{2}},\quad|h|^{2}\geq 0 \tag{6}\]  
where \(\lambda\) is the parameter of the exponential distribution. In this case, the SNR of the signal transmission from the source to the relay node can be expressed as follows:  
\[\text{SNR}_{1}=\frac{P_{s}C_{1}d_{1}^{-\alpha}|h_{sr}|^{2}}{n_{0}} \tag{7}\]  
The connection probability conditioned on \(\text{SNR}_{1}>\gamma_{0}\) is expressed as  
\[P_{y}(\text{SNR}_{1}>\gamma_{0})=P_{y}\bigg{\{}C_{1}d_{1}^{-\alpha}\frac{P_{s} |h_{sr}|^{2}}{n_{0}}>\gamma_{0}\bigg{\}} \tag{8}\]  
where \(P_{s}\) is the transmission power of the shore-based source, \(C_{1}\) is the path loss factor from the source to the relay node, \(d_{1}\) is the distance between the source and the relay node, \(\alpha\) is the path loss exponent, \(\gamma_{0}\) is the target SNR threshold, and \(n_{0}\) is the noise power. Assuming that the parameter \(\lambda\) is uniformly distributed within the interval \([a,b]\) to simulate offshore channel conditions under different scenarios, its distribution is given by  
\[f(\lambda)=\frac{1}{b-a},\quad\lambda\in[a,b] \tag{9}\]  
Substituting the above expression, the conditional probability expression can be obtained as follows:  
\[P_{y}(\text{SNR}_{1}>\gamma_{0})=\int_{a}^{b}\int_{\frac{\gamma_{0}n_{0}}{P_{ s}C_{1}d_{1}^{-\alpha}}}^{\infty}\frac{\lambda e^{-\lambda x}}{b-a}\,dx\,d\lambda \tag{10}\]By integrating with respect to \(|h|^{2}\) and \(\lambda\), the expression can be further simplified as  
\[P_{y}=\int_{a}^{b}\frac{1-F(x)}{b-a}\,d\lambda=\int_{a}^{b}\frac{e^{-\lambda\frac {\gamma_{0}\Psi_{0}}{B_{\zeta}C_{1}d_{1}^{-2}}}}{b-a}\,dx\,d\lambda\]  
\[=\frac{1}{b-a}\frac{P_{\zeta}C_{1}d_{1}^{-\alpha}}{\gamma_{0}n_{0}}\Bigg{(}e^{ -a\frac{\gamma_{0}\Psi_{0}}{B_{\zeta}C_{1}d_{1}^{-2}}}-e^{-b\frac{\gamma_{0} \Psi_{0}}{B_{\zeta}C_{1}d_{1}^{-2}}}\Bigg{)} \tag{11}\]  
where \(F(x)\) is the cumulative distribution function (CDF) of the received signal and is expressed as an exponential distribution: \(F(x)=1-e^{-\lambda x},(x>0)\). Similarly, the probability expression for the second hop, where \(\text{SNR}_{2}>\gamma_{0}\), can be derived as  
\[P_{y}(\text{SNR}_{2}>\gamma_{0})=\int_{a}^{b}\int_{\frac{\gamma_{0}\Psi_{0}}{ B_{\zeta}C_{2}d_{2}^{-2}}}^{\infty}\frac{\lambda e^{-\lambda x}}{b-a}\,dx\,d\lambda\]  
\[=\frac{1}{b-a}\frac{P_{r}C_{2}d_{2}^{-\alpha}}{\gamma_{0}n_{0}}\Bigg{(}e^{-a \frac{\gamma_{0}\Psi_{0}}{B_{\zeta}C_{2}d_{2}^{-2}}}-e^{-b\frac{\gamma_{0} \Psi_{0}}{B_{\zeta}C_{2}d_{2}^{-2}}}\Bigg{)} \tag{12}\]  
where \(P_{r}\) is the transmission power at the relay node, \(C_{2}\) is the path loss factor from the relay node to the receiving end, and \(d_{2}\) is the distance from the relay node to the target node. Relay cooperative communication techniques are primarily categorized into amplify-and-forward (AF) and decode-and-forward (DF) schemes based on their signal processing methods. In the AF mode, the relay nodes directly amplify and forward the noisy source signal without performing demodulation or decoding. Conversely, the DF mode involves the relay nodes demodulating and decoding the received signal first, followed by re-encoding and modulating it before its transmission to the destination node [18].