## 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 implementation of the method is as follows: for each fixed time value \(t\), the weights \(V_{k}\) and the values of the function \(W_{E}(s)\) at the points \(s=\frac{kln(2)}{t}\) are calculated. These are then used to compute a weighted sum, normalized by a multiplier of \(\frac{ln(2)}{t}\). By repeating this procedure for multiple values of \(t\) within a specified range, a discrete approximation of the function \(f_{T}(t)\) can be obtained. The resulting probability density function \(f_{T}(t)\) allows for not only for determining the expected request processing time but also for analyzing the probability of meeting specified deadlines, constructing confidence intervals for processing time, and assessing the risk of critical delays. The constructed transfer function (5) of the GERT model for the first stage of user request processing takes the form of a fractional-rational expression, as follows:  
\[W_{E}(s)=\frac{P(s)}{Q(s)} \tag{9}\]where \(P(s)\) and \(Q(s)\) are polynomials in the variable \(s\). To obtain the analytical expression of the probability density function \(f_{T}(t)\), it is necessary to perform the inverse Laplace transform of \(W_{E}(s)\). However, for fractional-rational functions, direct inversion is difficult. Therefore, a standard solution procedure is applied, as recommended in works [25]. The process of recovering the function \(f_{T}(t)\) consists of several key steps. In the first step, a change of variable is made, \(s=-z\), as a result of which the transfer function takes the following form:  
\[W_{E}(-z)=\frac{P(-z)}{Q(-z)} \tag{10}\]  
This substitution is used to facilitate the subsequent analysis of the function's singularities in the complex plane. After the variable change, the roots of the equation \(Q(-z)=0\) are located in the right half-plane, which simplifies their interpretation and makes the subsequent decomposition of the function into partial fractions easier. Using expression (5), we obtain the following equality:  
\[W_{E}(s)=\frac{-p_{1}^{2}p_{2}p_{3}\lambda_{2}\lambda_{3}(\frac{\lambda_{1}^{ 2}}{\lambda_{1}-z})^{k}(\lambda_{4}-z)(\lambda_{5}-z)}{(\lambda_{2}-z)( \lambda_{3}-z)((p_{1}\lambda_{1})^{k}p_{4}p_{5}\lambda_{4}\lambda_{5}-( \lambda_{4}-z)(\lambda_{5}-z)(\lambda_{1}-z)^{k})} \tag{11}\]  
The next step is solving the equation \(Q(-z)=0\). The roots of this equation \(z_{i}\) may be real numbers (real roots) or form pairs of complex conjugates. Each root can be either simple (multiplicity 1) or multiple (multiplicity \(m_{i}>1\)). Finding the roots of the denominator is a key step, as each root determines the behavior of the corresponding component in the result after the inverse transformation. Once all roots \(z_{i}\) of the denominator are found, the fraction is decomposed \(\frac{P(-z)}{Q(-z)}\) into a sum of simple elements. According to standard partial fraction theory, any rational expression can be represented as a sum of the following:  
\[W_{E}(-z)=\sum_{i}\sum_{k=1}^{m_{i}}\frac{A_{ik}}{(z-z_{i})^{k}} \tag{12}\]  
where \(m_{i}\) is the multiplicity of the root \(z_{i}\), and \(A_{ik}\) are the decomposition coefficients. The coefficients \(A_{ik}\) are determined using standard methods. This may involve comparing polynomial coefficients after bringing them to a common denominator, or using the method of equivalent residues at the poles (residue method). If the root \(z_{i}\) is simple (\(m_{i}=1\)), the corresponding term of the fraction has the form \(\frac{A_{ik}}{(z-z_{i})}\). If the root is multiple, all degrees from \((z-z_{i})^{-1}\) to \((z-z_{i})^{-m_{i}}\) are taken into account. In the next stage of the process of recovering the function \(f_{T}(t)\), known results of the inverse Laplace transform [24] can be used for each term of the decomposition.