## 3 Model Assumption and Toss-and-Catch Algorithm
### Score-Based Selection Mechanism
#### 3.3.2 Scoring and Ranking  
1. Scoring of OIS connection hold time (\(score_{\text{hold}}\))  
When the train increases its moving speed, the time that the OIS spends within the proximity of an SBS becomes shorter, possibly leading to more frequent handovers. If the OIS determines that a handover is necessary at \(t_{n+1}\), it would select a suitable SBS from SBSG1 for the handover. By selecting an SBS that allows for a longer connection hold time, we can reduce the frequency of handovers. With respect to the geometric relationship between the track equation and the SBS currently being scored, one of two possible scenarios occurs: (1) the track equation is tangent to the SBS; or (2) the track equation is secant to the SBS. Figure 11 illustrates these two scenarios for this geometric relationship. A score of 0 is assigned for Scenario 1; and as for Scenario 2, the scoring method is described below. The circle equation of SBS \(i\) can be expressed as [24]  
\[(x-X_{i})^{2}+(y-Y_{i})^{2}=d_{i}^{2}. \tag{12}\]  
Through (3) and (12), we can obtain the two cut point coordinates \((P_{\text{S1}x},P_{\text{S1}y})\) and \((P_{\text{S2}x},P_{\text{S2}y})\) of the track equation and SBS \(i\). They can be expressed as  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]  
\[(P_{\text{S2}x},P_{\text{S2}y})=(\frac{-B-\sqrt{B^{2}-4 AC}}{2A},m\frac{-B- \sqrt{B^{2}-4 AC}}{2A}+b), \tag{14}\]  
where  
\[A=m^{2}+1, \tag{15}\]  
Figure 11: Geometric relationship between the track equation and the SBS currently being scored. \[B=-2X_{i}+2 mb-2 mY_{i}, \tag{16}\] \[C={X_{i}}^{2}+b^{2}-2 bY_{i}+{Y_{i}}^{2}-{d_{i}}^{2}. \tag{17}\]  
We defined \(D_{\mathrm{secant}}\) as the secant distance between the two points \((P_{\mathrm{S1x}},P_{\mathrm{S1y}})\) and \((P_{\mathrm{S2x}},P_{\mathrm{S2y}})\), and expressed it as  
\[D_{\mathrm{secant}}=\sqrt{\left(\frac{\sqrt{B^{2}-4 AC}}{A}\right)^{2}-\left( \frac{m\sqrt{B^{2}-4 AC}}{A}\right)^{2}}. \tag{18}\]  
We denoted \(T_{\mathrm{hold}}\) as the OIS's connection hold time with the SBS and defined it as  
\[T_{\mathrm{hold}}=\frac{D_{\mathrm{secant}}}{v^{I_{n}}}. \tag{19}\]  
The connection hold time \(score_{\mathrm{hold}}\) was defined as  
\[score_{\mathrm{hold}}=\frac{T_{\mathrm{hold}}}{\frac{2d_{i}}{v^{I_{n}}}}\times 1 00. \tag{20}\]  
The track equation intersecting the SBS emission center indicates the maximum connection hold time that can be achieved; therefore, a \(score_{\mathrm{hold}}\) of 100 is given when this occurs. When the track deviates from the SBS emission center, connection hold time becomes shorter, and a lower score is given. For example, consider the two secant scenarios as shown in \[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]0 If the secant lengths are \(1.74d_{i}\) and \(2d_{i}\), the scores (as calculated via (20)) given are 87 and 100, respectively. (2) Scoring of maximum RSS available to the OIS (\(score_{\mathrm{RSSmax}}\))  
Signal quality is important in the communication process. By selecting an SBS that offers more RSS, the OIS is able to increase the average strength of the signals that it receives. With \(P_{\mathrm{RX}}\) denoting the OIS's received power, RSS is defined as [25]  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]1  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]2 Calculation of \(score_{\mathrm{hold}}\).