## 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
For a fraction of the form \(\frac{1}{(z-z_{i})}\) the inverse transform has the following form:  
\[W_{E}(s)=\frac{p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{2}}{ \lambda_{1}+s})(\lambda_{4}+s)(\lambda_{5}+s)}{(\lambda_{2}+s)(\lambda_{3}+s)( (p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-(\lambda_{4}+s)(\lambda_ {5}+s)(\lambda_{1}+s)^{k})} \tag{5}\]0  
For a fraction of the form \(\frac{1}{(z-z_{i})^{k}}\) (multiple root), as follows:  
\[W_{E}(s)=\frac{p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{2}}{ \lambda_{1}+s})(\lambda_{4}+s)(\lambda_{5}+s)}{(\lambda_{2}+s)(\lambda_{3}+s)( (p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-(\lambda_{4}+s)(\lambda_ {5}+s)(\lambda_{1}+s)^{k})} \tag{5}\]1  
Thus, each pole (root) contributes to the final distribution function in the form of an exponential, possibly multiplied by a power function of time. At the next stage, by summing the results of the inverse transform for all terms of the decomposition, we obtain the complete expression for the probability density function of request processing time, as follows:  
\[W_{E}(s)=\frac{p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{2}}{ \lambda_{1}+s})(\lambda_{4}+s)(\lambda_{5}+s)}{(\lambda_{2}+s)(\lambda_{3}+s)( (p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-(\lambda_{4}+s)(\lambda_ {5}+s)(\lambda_{1}+s)^{k})} \tag{5}\]2  
Each term represents an exponential function of time multiplied by a power function \(t^{k-1}\), weighted by the coefficient \(A_{ik}\). In the case where the roots of the denominator are complex conjugate numbers, their contribution to the function \(f_{T}(t)\) will be real. This is ensured by the fact that the sums of exponentials of conjugate roots result in cosine and sine functions via [PERSON]'s formulas. For example, if \(z_{i}=\alpha+i\beta\) and \(\widetilde{z}_{i}=\alpha-i\beta\), then the following occurs:  
\[W_{E}(s)=\frac{p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{2}}{ \lambda_{1}+s})(\lambda_{4}+s)(\lambda_{5}+s)}{(\lambda_{2}+s)(\lambda_{3}+s)( (p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-(\lambda_{4}+s)(\lambda_ {5}+s)(\lambda_{1}+s)^{k})} \tag{5}\]3  
and the contribution of the function will be expressed through damped oscillations. Let us conduct experimental studies of the presented mathematical model of the first stage of user request processing. We will determine the probability density function of the processing time for the first stage of user request handling. In doing so, we will define the following branch parameters of the GERT network as initial data: \(p_{1}=p_{3}=p_{5}=0.9999\), \(p_{2}=0.7\), \(p_{4}=0.3\), \(\lambda_{1}=1\), \(\lambda_{2}=0.9\), \(\lambda_{3}=0.8\), \(\lambda_{4}=0.85\), \(\lambda_{5}=0.91\), and \(k=2\). Substituting the specified values into Expression 4, we obtain the following form of the equation \(Q(-z)=0\):  
\[W_{E}(s)=\frac{p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{2}}{ \lambda_{1}+s})(\lambda_{4}+s)(\lambda_{5}+s)}{(\lambda_{2}+s)(\lambda_{3}+s)( (p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-(\lambda_{4}+s)(\lambda_ {5}+s)(\lambda_{1}+s)^{k})} \tag{5}\]4  
The real roots of this equation are \(z_{1}=0.2430\), \(z_{2}=0.8000\), \(z_{3}=0.9000\), and \(z_{4}=1.6369\). The complex conjugate roots of this equation are \(z_{5}=0.9401-0.6911i\), \(z_{6}=0.9401+0.6911i\). After finding the roots of the equation \(Q(-z)=0\), the next step is the numerical determination of the probabilistic characteristics of the request traversal time through the model of the first stage. For this, the inverse Laplace transform method is used, which allows obtaining the probability density function \(f_{T}(t)\) based on the transfer function of the GERT network. Based on the obtained density function, the key statistical parameters of the first stage were calculated. * Mathematical expectation: \(E[T]=\textbf{29.294}\);
* Standard deviation: \(\sigma=\textbf{0.707}\).