# End-to-End Delay Model for Train Messaging over Public Land Mobile Networks
## 6 Results
### Example of Model Application and Validation  
In this section, we provide an example of application of the proposed E2E model in a realistic railway scenario. The model is used to reproduce a situation similar to that depicted in Figure 3. For this purpose, we have considered one received power profile obtained from experimental data and we have first identified the handover points along the track. The E2E delay profile has been generated using the model described in Section 4. In particular, at each time instant integer multiple of \(\Delta T=200\) ms, we have generated the E2E delay and results have been plotted in Figure 11a,b corresponding to two samples of the E2E process. To obtain results, we have assumed \(p_{H}=1\) i.e., in the case of handover message, queueing always occurs. The random variables, \(X_{0}\), \(X_{H}\) and \(X_{1}\) have been generated numerically in accordance with the statistics in Figures 9 and 10, respectively. For simplicity, we have restricted the generation of the E2E delays to the case of one radio technology i.e., the WCDMA 2100; furthermore, no (sporadic) message loss has been considered. The generated E2E delay behavior can reproduce the main features observed in the experimental data in Figure 3. As expected, low received power levels usually determine the enqueue of messages. In order to validate the proposed model in Figure 12a,b, we compare the Cumulative Density Function (CDF) of the E2E delay from experimental data with the CDFs of the E2E delay obtained by running the proposed model. Results have been achieved considering the same railroad section as in Figure 11 in two different days. The two scenarios are characterized by (partially) different received power profiles along the track and partially different time instants of handover. For each scenario, we have run the proposed E2E delay model several times. The model has been driven by the power profile and  
Figure 11: Example of E2E delay generation based on a given received power profile and for a single technology (i.e., WCDMA 2100). handover positions obtained from the corresponding experimental data. To generate the E2E delays, we have used the PDFs of \(X_{0}\), \(X_{1}\) and \(X_{H}\) indicated in the previous section. Results obtained from the model (curves with different colors) show a very good agreement with experimental data (black curve) i.e., the CDFs of E2E delay obtained from model are close to the corresponding experimental CDF. Negligible discrepancies obtained for very small E2E delays can be due to statistical instabilities of the PDF of \(X_{0}\). In Figure 13, we plot the CDF of the E2E delays obtained from the model after having collected all the E2E delays generated in each sample. Results in the figure make evident the good agreement between experimental and model-based results even for large E2E delays (e.g., up to seconds). We have repeated validation in the railroad section corresponding to the time interval 2.4 h-3.2 h in Figure 3. This part of the rail track is characterized by handovers/roaming mainly between WCDMA 2100 and WCDMA 900. The statistics of the random variables \(X_{0}\), \(X_{1}\) and \(X_{H}\) used to generate the E2E delays are those in Figures 9 and 10. Figure 12: Comparison of E2E delay Cumulative Density Function generated from the model with the corresponding experimental Cumulative Density Function: Two different measured samples on the same railway section (WCDMA 2100 technology only). Figure 13: Comparison of E2E delay CDF generated from the model after collecting E2E delays of each generated samples: Expanded time scale up to 2 s, two different experimental samples of the same railway section (WCDMA 2100 technology only). In Figure 14, we compare the CDFs of the E2E delays generated by the model (curves with different colors) with those obtained from experimental data (black curve) on the same rail track and corresponding to two different days. All available radio technologies are considered.