## 3 Model Assumption and Toss-and-Catch Algorithm
### Gathering of Information about the Route Ahead and the SBSs Deployed around This Route  
#### 3.1.1 Track Path Calculation and Coordinate Conversion  
As shown in Figure 4, a kernel is first defined before calculating the track path. This kernel is a square with a side length of 10 coordinate units, with the length from the center to the vertex of each 1 \(\times\) 1 small square being defined as the preset signal coverage radius \(R\) of the actual SBS, and this describes the relationship between the kernel coordinates and their corresponding actual geographic coordinates. Using trigonometric functions, it can be deduced that the ratio of the side length of each 1 \(\times\) 1 square to the actual geographic length is one coordinate unit: \(\sqrt{2}R\) m. Each time the OIS obtains information relating to the topologic conditions ahead of the train, it must select a kernel block (with an actual geographic area of \(10\sqrt{2}R\times 10\sqrt{2}R\) m\({}^{2}\)) and obtain information on the block (SBSs' coordinates, load, coverage, and train track path). Figure 5 presents a Google Map showing the Taiwan high speed rail (THSR) Taipei Station and the THSR Taoyuan Station, providing an example that illustrates how kernel block selection is implemented on an actual map. In this figure, the blue solid line plots the track route from the THSR Taipei Station to the THSR Taoyuan Station, while the black bordered boxes are the kernels. A close-up of one of the kernels reveals the schematic relationship between the track and the SBSs after coordinate conversion (the conversion to the kernel coordinate system is described in Appendix A). Figure 4: Kernel coordinate system model. Figure 3: Flowchart of the proposed toss-and-catch algorithm. As the overall curvature of the track is not significant and the geographic coordinate range that is selected each time represents only a very small segment of the track, the track curvature within each selected block range is in fact very small. To facilitate modeling and calculations, we approximated each selected track path using a linear equation with two unknowns, which is given by [24]  
\[y=mx+b, \tag{3}\]  
where \(m\) is the slope and \(b\) is the constant term. \(m\) and \(b\) can be obtained by calculating the two intersection points, \(P_{\text{entry}}(X_{\text{entry}},Y_{\text{entry}})\) and \(P_{\text{exit}}(X_{\text{exit}},Y_{\text{exit}})\), of the track and the boundary of the selected block range and bringing in (3). The distance between \(P_{\text{entry}}(X_{\text{entry}},Y_{\text{entry}})\) and \(P_{\text{exit}}(X_{\text{exit}},Y_{\text{exit}})\) can be expressed as [24]  
\[D_{\text{rail}}(P_{\text{entry}},P_{\text{exit}})=\sqrt{(X_{\text{entry}}-X_{ \text{exit}})^{2}+(Y_{\text{entry}}-Y_{\text{exit}})^{2}}. \tag{4}\]  
A range is always selected randomly by the kernel, and consequently, the coordinate point of the OIS (namely, \(P_{\text{entry}}(X_{\text{entry}},Y_{\text{entry}})\)) when it enters the kernel range is random. To ensure that each selected track is sufficiently long, we defined \(D_{\text{rail}}(P_{\text{entry}},P_{\text{exit}})\) to be valid only when it is greater than or equal to 90% of the kernel's side length, as shown in Figure 6; when \(D_{\text{rail}}(P_{\text{entry}},P_{\text{exit}})\) is less than 90% of the kernel's side length, the selection is repeated until a valid length is achieved for \(D_{\text{rail}}(P_{\text{entry}},P_{\text{exit}})\). It should be noted that whenever the OIS leaves the current selected range (with the exit point being \(P_{\text{exit}}(X_{\text{exit}},Y_{\text{exit}})\)), the  
Figure 5: Schematic diagram showing the kernel block selection range in relation to the actual geographic region. selection process is repeated on the path ahead of the train to continuously obtain track and topology information until the train arrives at its destination.