# End-to-End Delay Model for Train Messaging over Public Land Mobile Networks
## 3 Analysis of Experimental Data  
The analysis of experimental data is the first step required to provide evidence of the main features characterizing the behavior of the E2E delay over the rail line. The resulting E2E delay model should reproduce the features emerging from this analysis. In Figure 3, we plot one sample of the E2E delay as a function of the travel time as obtained during the experimental campaign. On to the same picture, we have superimposed the corresponding received power profile associated with the transmitted messages (one message every 200 ms in this case). In the same picture, we have also indicated the serving radio technology connecting the terminal along the railway path. At the moment of the measurement campaign, LTE was not present in the considered sub-urban/rural areas. Anyway, it is expected that LTE is and will be mainly deployed in high-densely populated areas. Thus, for regional lines, it is more appropriate to consider 2G and 3G radio technologies. The presence of the tunnel has been made evident in the picture. At the time of measurements, the 6.5 km long gallery was not equipped with any radio technology and roaming with GSM 900 technology was not performed until the train exits the gallery. In the following part of this section, we proceed with the examination of the measured E2E delays in order to assess the main features to be owned by the model to reproduce behavior similar to that in Figure 3.  
In the G state, the E2E delay is (practically) bounded within hundreds of milliseconds (see circled areas in Figure 4b), sometimes taking values higher than the messages generation interval (e.g., 200 ms), thus causing enqueuing of messages and then a transition in the state Q (see below). In Table 1, we report a sample of the E2E delays obtained from measurements representing the behavior in state G. The E2E delays and the corresponding transmission and receiving timestamps have been also indicated. Figure 3b,c are two zooms of Figure 3a. From results in Figure 3b,c, we can observe a correlation between the received power level and the experienced E2E delays. This relation will be studied in the section concerning the parameters of the E2E delay model presented in this paper. The typical pattern of the E2E delay profile obtained from experimental data is shown in Figure 4a. The plot in Figure 4b is a zoom of Figure 4a. Figure 4a provides evidence of a two states-like behavior for the E2E delay. Thus, we can introduce two distinct macro-states that, for convenience, we label as Good (G) and enQueued (Q). Figure 3: Example of the received power (left \(y-\)axis) and E2E delay (right \(y-\)axis) as a function of the travel time in (**a**). Corresponding zoomings are in (**b**,**c**). The red line indicates the duration of the time interval the train uses the radio technology as in the corresponding label (i.e., Wideband Code Division Multiple Access (WCDMA) 2100, WCDMA 900, GSM 900) for train to ground communications; the black line is the received power level measured by the on board terminal; the blue line is the measured E2E delay. From data in Table 1, in some cases (see numbers in bold), the receive time of the message can be greater than the transmit time of the successive message. This event can be explained assuming messages have been buffered in the transmitter when the link becomes (temporarily) unavailable and, more in general, due to buffering in any part of the traversed network sections (e.g., Radio Access Network (RAN), backbone, core). In the case of buffering in the transmitter, for messages of constant length, the speed for emptying the buffer depends on the assigned channel bit rate over the wireless link when it returns as available. In general, the instantaneous link bit rate can be considered as random, and its value depends on the locally available radio technology(ies) (e.g., 2G, 3G or 4G), on the traffic load in the cell and on the scheduling strategy of the serving base station (BS).