## 2 Materials and Methods
### GERT Model for User Data Processing in the Recommendation System
#### 2.2.1. GERT Model of the First Stage of User Request Processing
The conducted analysis of the process formalized in Figure 1 made it possible to establish the characteristics of the branches considered in the GERT model, as well as the distribution parameters. These are presented in more detail in Table 3.  
By substituting the expressions for each arc into expression (3), we obtain the following:  
\[W_{E}(s)=\frac{(p_{1}^{2}p_{2}p_{3})\times(\lambda_{1})^{2k}\lambda_{2}\lambda _{3}}{(\lambda_{1}+s)^{2k}(\lambda_{2}+s)(\lambda_{3}+s)}\times\frac{1}{1-p_{ 4}p_{5}p_{1}^{k}\times(\lambda_{4}+s)(\lambda_{5}+s)(\lambda_{1}+s)^{k}\lambda _{4}\lambda_{5}\lambda_{1}^{k}} \tag{4}\]  
After performing straightforward mathematical transformations, we arrive at the expression for calculating the equivalent \(W\)-function. \begin{table}
\begin{tabular}{c c c} \hline \hline
**W-Function** & \
\begin{tabular}{c} **Derivative of the Probability** \\ **Function** \\ \end{tabular} &
\begin{tabular}{c} **Derivative of the** \\ **Moment-Generating Function** \\ \end{tabular} \\ \hline \(W_{12}\) & \(p_{1}\) & \((\frac{\lambda_{1}}{\lambda_{1-s}})^{k}\) \\ \(W_{23}\) & \(p_{1}\) & \((\frac{\lambda_{1}}{\lambda_{1-s}})^{k}\) \\ \(W_{35}\) & \(p_{2}\) & \(\frac{\lambda_{2}}{\lambda_{2-s}}\) \\ \(W_{56}\) & \(p_{3}\) & \(\frac{\lambda_{2}}{\lambda_{2-s}}\) \\ \(W_{34}\) & \(p_{4}\) & \(\frac{\lambda_{4}}{\lambda_{4-s}}\) \\ \(W_{42}\) & \(p_{5}\) & \(\frac{\lambda_{5}}{\lambda_{5-s}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the industries considered in the GERT model, as well as the parameters of their distribution. \[W_{E}(s)=\frac{p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{2}}{ \lambda_{1}+s})(\lambda_{4}+s)(\lambda_{5}+s)}{(\lambda_{2}+s)(\lambda_{3}+s)( (p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-(\lambda_{4}+s)(\lambda_ {5}+s)(\lambda_{1}+s)^{k})} \tag{5}\]  
The resulting transfer function of the entire network \(W_{E}(s)\) represents the characteristic function of the request processing time. It is the Laplace transform of the desired probability density function of the processing time \(f_{T}(t)\). Formally, the relationship between the transfer function and the time distribution function is expressed as follows:  
\[\Phi_{T}(s)=W_{E}(s),\]  
where \(\Phi_{T}(s)\) is the characteristic function of the time. The probability density function \(f_{T}(t)\) can be obtained from \(W_{E}(s)\) through the inverse Laplace transform. Theoretically, this transformation is written in the form of the Bromwich integral, as follows:  
\[f_{T}(t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{it}W_{E}(s )ds \tag{6}\]  
where the integration is carried out along a vertical line in the complex plane, located to the right of all singularities of the function \(W_{E}(s)\). In practice, analytical computation of the Bromwich integral for complex transfer functions, such as our model, is challenging. Therefore, for the numerical recovery of the time distribution function, it is reasonable to use specialized numerical methods. In this study, the Stehfest method [32] was used to solve the inversion problem, as recommended in works on the theory of GERT networks [1; 33]. The Stehfest method allows the function \(f_{T}(t)\) to be approximated through a finite sum of transfer function values at various arguments. The following approximate formula is used:  
\[f_{T}(t)\approx\frac{ln(2)}{t}{\sum_{k=1}^{N}V_{k}W_{E}\frac{kln(2)}{t}} \tag{7}\]  
where \(N\) is an even number that determines the approximation accuracy (typically \(N\) = 10, 12, and 14), and \(V_{k}\) is the special Stehfest weighting coefficient. These coefficients are calculated using binomial coefficients according to the following formula:  
\[V_{k}={\sum_{j=2k+1}^{min(k,\frac{N}{2})}{\frac{j^{\frac{N}{2}}(\frac{N}{j})( j)}{(k-1)}(\frac{j}{\frac{j}{2}-j})}} \tag{8}\]  
where \((^{t}_{b})\) denotes binomial coefficients.