## 3 Mathematical Model of Radar Operation Under Interference Conditions
The equivalent W-function of the radar functioning process in interaction with a tracked object at the initial stage can be expressed as:  
\[W_{E}(\mathrm{s})=\frac{W_{12}W_{23}W_{34}W_{46}W_{67}+W_{12}W_{23}W_{34}W_{45 }W_{54}W_{46}W_{67}}{1-W_{12}W_{21}-W_{12}W_{23}W_{34}W_{46}W_{34}W_{61}} \tag{2}\]  
Let us derive an analytical relationship for calculating and studying the equivalent W-function of the radar operation process in its interaction with the tracking object during the initial stage. To describe the characteristics of the branches of the GERT network in the radar operation process when interacting with the tracking object, we utilize the moment-generating function of the exponential distribution. This choice is justified by several key aspects:  
* Analytical Convenience. The moment-generating function allows for straightforward manipulation of distributions when branches of the network are connected or processed in other ways. This is particularly beneficial for the exponential distribution, given the simplicity of its moment-generating function. * Addition and Parallelism. In GERT networks, it is often necessary to analyze serial and parallel connections of branches. The moment-generating function for the exponential distribution provides a simple means to analyze such configurations. Based on expression (2) and our research findings, we present the characteristics of the branches and the distribution parameters in Table 1.  
Table 1 describes the key characteristics of the branches in a GERT network used to model the interaction process between the radar system and the tracking object at the initial stage. Each branch has a W-function, which accounts for both the transition probability and the timing characteristics through moment generation functions. In GERT networks, it is common to work with sequential and parallel branch connections, and for such operations, the exponential distribution provides particularly convenient formulas for calculating overall characteristics. This aspect is crucial for formalizing radar system operation processes, as the time periods before interaction with the object or task completion may vary, and the exponential distribution is ideal for describing them. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{\lambda}\) & **Branch** & \(\mathbf{W}\)-Function & **Probability** & \multicolumn{1}{c}{**Moment Generating Function**} \\ \hline
1 & (1–2) & \(W_{12}\) & \(p_{1}\) & \(\lambda_{1}/(\lambda_{1}-s)\) \\
2 & (2–3) & \(W_{23}\) & \(p_{2}\) & \(\lambda_{2}/(\lambda_{2}-s)\) \\
3 & (2–1) & \(W_{21}\) & \(1-p_{2}=q_{1}\) & \(\lambda_{3}/(\lambda_{3}-s)\) \\
4 & (3–4) & \(W_{34}\) & \(p_{3}\) & \(\lambda_{4}/(\lambda_{4}-s)\) \\
5 & (4–5) & \(W_{45}\) & \(p_{4}\) & \(\lambda_{4}/(\lambda_{4}-s)\) \\
6 & (5–4) & \(W_{54}\) & \(p_{4}\) & \(\lambda_{5}/(\lambda_{5}-s)\) \\
7 & (4–6) & \(W_{46}\) & \(1-p_{4}=q_{2}\) & \(\lambda_{6}/(\lambda_{6}-s)\) \\
8 & (6–7) & \(W_{67}\) & \(p_{5}\) & \(\lambda_{7}/(\lambda_{7}-s)\) \\
9 & (6–1) & \(W_{61}\) & \(1-p_{5}=q_{3}\) & \(\lambda_{8}/(\lambda_{8}-s)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the functioning model branches process of the radar in interaction with the tracking object at the initial stage. For the presented characteristics of the model branches, we will estimate the equivalent W-function of the radar operation process during its interaction with the tracking object at the initial stage. Let us describe the W-function for each branch as follows:  
\[W_{ij}=p_{k}*\lambda_{m}, \tag{3}\]  
where \(i,j\) are node numbers in the GERT network;  
\(k\) is the probability index for the corresponding transition within the GERT network;  
\(m\) is the generating function index of the corresponding transition moments within the GERT network. \(\lambda\) is a key characteristic in the GERT model for describing transitions between different network states. It characterizes the rate of transitions or the interaction intensity of the network branch with the target under surveillance.