# End-to-End Delay Model for Train Messaging over Public Land Mobile Networks
## 6 Results
### Statistics of the Random Variables \(X_{0}\), \(X_{1}\) and \(X_{h}\)  
For each trial, we have collected the values of the E2E delay for each one of the transmitted and received messages. In the selected tracks, we had two tunnels having lengths of about 600 m and 6.5 km. It has been observed the shortest tunnel didn't cause any signal loss. At the time of the trial, the longest tunnel was not served by any radio technology. Data corresponding to message generation rate of 5 msg/s have been considered to derive the statistics of \(X_{0}\), \(X_{H}\) and \(X_{1}\). Furthermore, the (mean) received power \(P_{i}\) and the indication of the serving radio technology are available for each \(R_{i}\). For data analysis and classification of statistics, the original experimental data set of the E2E delays have been initially partitioned into three sub-sets on the basis of the serving radio technology: WCDMA 2100, WCDMA 900 and GSM 900 (see Figure 3). Then, E2E delays in each sub-set have been further partitioned by grouping data into the following four received power level intervals: \(P_{i}>-65\) dBm, \(-80\) dBm\(<P_{i}\leq-65\) dBm, \(-95\) dBm \(<P_{i}\leq-80\) dBm and \(P_{i}\leq-95\) dBm. For each group in each sub-set, state transition from G to Q have been identified using the condition \(R_{i-1}>T_{i}\) (see Figure 6). E2E delays measured during transitions have been collected to determine the statistics of \(X_{1}\). To obtain the statistics of \(X_{H}\) in each data group, we have also identified the time instants corresponding to handover events and we have collected the corresponding E2E delays. Handovers (soft or hard) are short time events and occur only a few times, especially when the train is moving on rural/open areas where base stations have large area coverages. We have identified handover events searching for the time instants where the received power inverts its tendency i.e., passing from (monotonically) decrease to (monotonically) increase. Some examples are indicated in Figure 3. In some cases during handover/roaming events, we have observed no enqueue of messages. This fact has been accounted for by the probability \(p_{H}\) in the model. Unfortunately, due to lack of information on the exact handover/roaming time, estimation of \(p_{H}\) is not accurate. In Figure 9, we plot the Probability Density Functions (PDFs) of the random variables \(X_{0}\) and \(X_{H}\) obtained from experimental data that have been ranked by technology and the received power interval. From Figure 9, the PDF of \(X_{0}\) is centered about some hundreds of milliseconds and its tails tend to increase with the lowering of the received power i.e., in the low power case the probability of having larger E2E delays increases and this could be due to the reduced transmission rate experienced by the terminal. This proves the (slight) correlation between E2E statistics and the received power level. Larger E2E delays have been observed in the GSM 900 technology case. This is due to the lower data rate provided by GSM with respect to WCDMA 2100 and WCDMA 900. To better make evident the (non-zero) probability of achieving E2E delays of tens of seconds, we have used the log-log scale in  
Figure 9: Probability Density Functions of the random variables \(X_{0}\) and \(X_{H}\) obtained from experimental data ranked by technology and the received power interval: \(X_{0}\) and \(X_{H}\) for WCDMA 2100 (**a,b**); \(X_{0}\) and \(X_{H}\) for WCDMA 900 (**c,d**); \(X_{0}\) and \(X_{H}\) for GSM 900 (**e,f**). the graphs in Figure 9. Fortunately, these events are rare and this causes the unstable behavior of the distribution tails. The dependence on the received power level can even be observed in the \(X_{H}\) statistics. E2E delays due to handover events are on the order of some hundreds of ms with high probability, especially in the case of WCDMA 2100 and WCDMA 900 where soft-handover occurs. Instead, as expected, the probability of recording E2E delays larger than 1 s is higher in the GSM 900 case.