## 2 Materials and Methods
### GERT Model for User Data Processing in the Recommendation System
#### 2.2.2 GERT Model of the Stage for Assigning Quantitative Weights to Identified Categories
Other arcs correspond to actions typically performed in one step or within the framework of simple administrative logic (for example, revalidation, error return, and processing completion). For them, the exponential distribution was selected, reflecting the memoryless (Markovian) nature of processing [33]. Thus, the model accounts for the varying temporal complexity of different stages, ensuring more realistic behavior of the processing time function at this stage. The process of restoring the probability density function of processing time \(f_{T}(t)\) from the transfer function of the GERT network \(W_{E}(s)\) begins with a transition to the complex frequency domain by substituting the variable \(s=-z\). The transfer function then takes the following form:  
\[W_{E}(-z)=W_{12}(-z)W_{23}(-z)\frac{W_{35}(-z)}{1-W_{34}(-z)W_{42}(-z)W_{23}(-z )-W_{52}(-z)W_{23}(-z)} \tag{18}\]  
where each component function \(W_{ij}(-z)\) is calculated taking into account the variable substitution and the corresponding form of the Laplace transform. In this GERT model, the denominator has the following structure:  
\[Q(z)=(\lambda_{1}-z)^{k}(\lambda_{2}-z)^{k}(\lambda_{3}-z)D(z) \tag{19}\]  
where \(D(z)\) is the resulting expression, as follows:  
\[D(z)=1-(\frac{p_{4}\lambda_{4}}{\lambda_{4}-z}\frac{p_{5}\lambda_{5}}{\lambda_ {5}-z}(\frac{p_{2}\lambda_{2}}{\lambda_{2}-z})^{k})-(\frac{p_{6}\lambda_{6}}{ \lambda_{6}-z}(\frac{p_{2}\lambda_{2}}{\lambda_{2}-z})^{k}) \tag{20}\]  
Thus, the poles \(z_{i}\) are defined as the solutions to the following equation:  
\[Q(z)=0\]  
The roots found \(z_{i}\in\) (real or complex conjugate) determine the behavior of the function \(f_{T}(t)\), its shape, symmetry, and the \"tail\" of the distribution, as well as the position of the expected value and the nature of the exceedance probability. \begin{table}
\begin{tabular}{c c c} \hline
**W-Function** & **Derivative of the Probability** & **Derivative of the** \\  & **Function** & **Moment-Generating Function** \\ \hline \(W_{12}\) & \(p_{1}\) & \((\frac{\lambda_{1}}{\lambda_{1}-z})^{k}\) \\ \(W_{23}\) & \(p_{2}\) & \((\frac{\lambda_{2}}{\lambda_{2}-z})^{k}\) \\ \(W_{35}\) & \(p_{3}\) & \(\frac{\lambda_{3}}{\lambda_{3}-z}\) \\ \(W_{34}\) & \(p_{4}\) & \(\frac{\lambda_{4}}{\lambda_{4}-z}\) \\ \(W_{42}\) & \(p_{5}\) & \(\frac{\lambda_{5}}{\lambda_{5}-z}\) \\ \(W_{52}\) & \(p_{6}\) & \(\frac{\lambda_{6}}{\lambda_{6}-z}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Characteristics of the industries considered in the GERT model and distribution parameters. The procedure for computing the roots is similar to the first example (model 12-16) and consists of the following steps. 1. The function \(W_{E}(s)\) is formed using all parameters \(p_{i}\), \(\lambda_{i}\) with specific values substituted. 2. For each time value \(t\in[0,T_{max}]\), an approximate value of \(f_{T}(t)\) is calculated. 3. Based on the array of \(f_{T}(t)\), the following are constructed: * the density function; * the distribution function \(F_{T}(t)\int_{0}^{t}f_{T}(\tau)d\tau\); * the exceedance probability \(1-F_{T}(t)\); * the mathematical expectation and variance: \[E[T]=\int_{0}^{\infty}tf_{T}(t)dt\] \[Var(T)=\int_{0}^{\infty}(t-E[T])^{2}f_{T}(t)dt\]  
After substituting the specified model parameters and performing the change \(s=-z\), the characteristic equation of the GERT network for the second stage takes the following form:  
\[Q(-z)=0.3019z^{8}-2.3246z^{7}+7.4991z^{6}-13.0734z^{5}+13.2067z^{4}-7.6483z^{3} +2.3171z^{2}-0.2878z+0.0096\]  
The following parameters were used in the calculations: \(p_{1}=p_{2}=0.95\), \(p_{3}=0.6\), \(p_{4}=0.5\), \(p_{6}=0.3\), \(\lambda_{1}=1.4\), \(\lambda_{2}=1.3\), \(\lambda_{3}=1.4\), \(\lambda_{4}=0.3\), \(\lambda_{5}=0.5\), \(\lambda_{6}=0.1\), and \(k=2\). For numerical analysis of the dynamics of time characteristics, it is necessary to determine the roots of the equation \(Q(-z)=0\). The results of solving this polynomial are presented below. Real roots: \(z_{1}=1.4657\), \(z_{2}=1.3998\), \(z_{3}=0.1813\), \(z_{4}=0.0515\).