## 3 Mathematical Model of Radar Operation Under Interference Conditions
Since a GERT network represents a random graph network, \(\lambda\) in each transition reflects the probabilistic time required to move between the network nodes:  
* \(\lambda_{1}\) describes the detection rate of the radar system. The larger the value of \(\lambda_{1}\), the faster the detection and transition to the next stage occur;
* \(\lambda_{2}\) characterizes the intensity of interaction between nodes 2 and 3, related to signal processing or detection confirmation. This parameter depends on factors such as signal quality and noise level;
* \(\lambda_{3}\) describes the probability of returning from node 2 to node 1. The value of \(\lambda_{3}\) determines how often a return to the previous stage is required;
* \(\lambda_{4}\) describes the transition intensity from node 3 to node 4, which may correspond to the initiation of target tracking. This time depends on radar parameters and target characteristics;
* \(\lambda_{4}\) and \(\lambda_{5}\) determine the transition intensity between the stages of track refinement and the need to return to previous stages for trajectory clarification;
* \(\lambda_{6}\) characterizes the probability of a successful transition to the next stage of interaction, which is associated with completing the target tracking process and transferring control to the next node;
* \(\lambda_{7}\) reflects the successful completion of the tracking process and the transition to the next stage, such as target confirmation;
* \(\lambda_{8}\) is associated with returning to the beginning of the cycle, which may be caused by the need to revalidate data or detect new objects. In this article, it is impossible to fully formalize and describe each of the presented parameters, \(\lambda_{m}\). This will be the focus of further research and mathematical formalization. Using expression (3) we obtain:  
\[W_{\text{E}}(s)=\frac{\frac{p_{1}p_{2}p_{3}p_{4}\lambda_{1}\lambda_{2}\lambda_{4 }\lambda_{6}\lambda_{7}\left((\lambda_{5}-s)^{2}+p_{4}^{2}\lambda_{5}^{2} \right)}{(\lambda_{1}-s)(\lambda_{2}-s)(\lambda_{4}-s)(\lambda_{5}-s)^{2}( \lambda_{6}-s)(\lambda_{7}-s)}}{\left(p_{1}q_{1}\lambda_{1}\lambda_{3}(\lambda _{2}-s)(\lambda_{4}-s)(\lambda_{6}-s)(\lambda_{8}-s)\right)}} \tag{4}\]  
Through standard mathematical transformations, the following expression can be obtained for calculating the equivalent \(W\)-function of the radar operation process in interaction with the tracking object at the initial stage:  
\[W_{\text{E}}(s)=\frac{r(s^{4}-ys^{3}+us^{2}-xs+c)}{(\lambda_{7}-s)(s^{7}+vs^{ 6}+bs^{5}+as^{4}+ds^{3}+fs^{2}+gs^{5}+h)}, \tag{5}\]  
where \(r=p_{1}p_{2}p_{3}p_{5}q_{2}\lambda_{1}\lambda_{2}\lambda_{4}\lambda_{6}\lambda _{7}\);  
\[w_{5}=(\lambda_{7}b-a);\] \[w_{4}=(\lambda_{7}a-d);\] \[w_{3}=(\lambda_{7}d-f);\] \[w_{2}=(\lambda_{7}f-g);\] \[w_{1}=(\lambda_{7}g-h);\] \[w_{0}=(\lambda_{7}h).\]  
Thus, the distribution density of the \"interaction\" time with the tracking object is:  
\[ \begin{array}{l}\varphi(x)=\sum\limits_{k=1}^{8}\text{Res}[\text{e}^{2\kappa }\Phi(z)]=\sum\limits_{k=1}^{8}\frac{\mu(z_{k})}{\psi(z_{k})}=\\ =\sum\limits_{k=1}^{8}\frac{e^{zx}(z^{4}-yz^{3}+uz^{2}-xz+c)}{8z^{7}-7w_{7}z^{6 }-6w_{6}z^{5}-5w_{5}z^{4}-4w_{4}z^{3}-3w_{3}z^{2}-2w_{2}z-w_{1}}\end{array}  \tag{9}\]  
The function may have not only simple poles, determined by the roots of the equation  
\[z^{8}-w_{7}z^{7}-w_{6}z^{6}-w_{5}z^{5}-w_{4}z^{4}-w_{3}z^{3}-w_{2}z^{2}-w_{1}z +w_{0}=0, \tag{10}\]  
but also poles of higher order. This happens when \(\lambda_{7}\) equals \(z\). In such cases, the distribution density of the \"interaction\" time with the tracking object is calculated using the expression for finding residues at the poles of the corresponding order. Thus, a mathematical model of radar operation under jamming conditions was developed based on the exponential GERT network.