## 3 Results  
_Numerical Analysis of the Time Characteristics of User Request Processing Stages and the Integrated Model_  
Through numerical analysis and study of the GERT models for the stages of user request processing and the integrated model, a series of graphs was constructed to enable a comprehensive analysis of the system's temporal behavior. The calculations used parameters obtained through structural analysis of the network, including transition probabilities along the arcs, processing intensities, and the depth of categorical analysis. As part of the numerical analysis of the GERT models describing the sequential stages of user request processing, a visualization of the time characteristics of each individual stage, as well as their combined effect within the integrated model, was performed. Based on the transfer functions constructed using parameters of transition probabilities, processing intensities, and the depth of categorical analysis, probability density functions of execution time, distribution functions, and exceedance probabilities of a specified time threshold were calculated. Figure 7: Sensitivity of stage 2 output to parameter variation. Figure 8 presents the graphs of the probability density functions of execution time for the respective stages and the integrated model as a whole. Probability density plots of execution time for the user request processing stages and the integrated model as a whole (Figure 8). On the probability density function graph \(f_{T}(t)\) (Figure 8), distinct differences between the time profiles of individual stages are visually noticeable. Stage 3 shows the steepest decline in density, indicating a greater concentration of probabilities near shorter time values. This stage completes more quickly compared to the others. Stages 1 and 2 are characterized by more \"stretched\" curves, reflecting greater variability in their execution time. The full model (shown by the bold red line) demonstrates averaged behavior, smoothed due to the composition of all three processes, with lower density near the start and a broader \"tail\". Figure 9 presents the distribution function graphs of execution time for the stages under consideration and for the integrated model as a whole. Figure 8: Probability density plots of execution time for the user request processing stages and the integrated model as a whole. Figure 9: Distribution function graphs of execution time for user request processing and the integrated model as a whole. The distribution function \(F_{T}(t)\) (Figure 9) confirms the previously drawn conclusions: for Stage 3, the value of \(F_{T}(t)\) reaches the 0.95 level faster than for Stages 1 and 2. This means that with 95% probability, the third stage will be completed faster than the other two. For the full model, the curve FT(t) lies below the others throughout the entire time interval, reflecting the overall inertial effect, indicating that the resulting behavior is slower due to the accumulated impact of all stages. Figure 10 presents the graphs of the exceedance probability of execution time for user request processing and the integrated model. The graph of the exceedance probability of execution time \(1-F_{T}(t)\) (Figure 10), presented on a logarithmic scale, clearly demonstrates the exponential decline in the probability of process incompletion as time increases. The highest exceedance probability values are observed for the full model, especially in the interval \(t=30\div 33\), indicating the overall effect of \"accumulated uncertainty\" when combining several sequential stages. Statistical processing of the results made it possible to obtain values of the mathematical expectation and standard deviation for each stage and the full model. The average execution time of the stages ranges from 29.00 (Stage 3) to 29.71 (Stage 2), while for the full model it reaches 29.80. The standard deviation across all models is approximately 2.2-2.3, indicating a comparable level of value dispersion.