## 3 Model Assumption and Toss-and-Catch Algorithm
### Gathering of Information about the Route Ahead and the SBSs Deployed around This Route
#### 3.1.2 SBS Categorization and Mode Setting  
The SBSs within the kernel range (collectively represented by \(\Re\)) are first sorted into two categories, namely MSBSs (collectively represented by \(\Re_{\text{MSBS}}\)) and non-MSBSs (collectively represented by \(\Re_{\text{non-MSBS}}\)). The nearby SBSs that the OIS has a high chance of coming into contact with are marked as MSBSs. In this marking process, we defined \((X_{i},Y_{i})\) as the coordinates of SBS \(i\) in the kernel coordinate system, and the shortest distance \(L_{i}\) between each SBS (within \(\Re\)) and the track is expressed as [24]  
\[L_{i}=\frac{|mX_{i}-Y_{i}+b|}{\sqrt{m^{2}+1}},\ i\in\Re. \tag{5}\]  
An SBS is marked as an MSBS when \(L_{i}\leq\sqrt{2}/2\), and as a non-MSBS when \(L_{i}>\sqrt{2}/2\). This criterion is visualized as a conceptual diagram in Figure 7. The rationale for establishing this criterion is as follows: because \(R\) (i.e., boot preset mode) is the preset coverage radius of an SBS in the actual geographic coordinate system, the OIS must be within the coverage of this SBS when the shortest distance between this SBS and the track is within \(R\). Moreover, the conversion of the actual geographic coordinate system to the kernel coordinate system shows that length \(R\) in the actual geographic coordinate system is equivalent to length \(\sqrt{2}/2\) in the kernel coordinate system. It should be noted that the modes of these MSBSs are subsequently set to super mode to maximize signal coverage in an attempt to cover the entire track. After an OIS exits the kernel range, SBS \(i\) within \(\Re_{\text{MSBS}}\) (in the range) is switched to the appropriate modes based on \(\gamma_{i}\) (i.e., the adjustment is based on (2)). Figure 6: Schematic diagram showing four scenarios in which the OIS uses a kernel to select the track range, with Scenarios 1 and 2 being invalid selections and Scenarios 3 and 4 being valid selections. After marking the MSBSs, the next step is to subcategorize the non-MSBSs. When an OIS requires a handover but it is unable to identify a suitable MSBS or there are no available MSBSs within the vicinity, it then needs to select suitable auxiliary SBSs from the \(\Re_{\text{non-MSBS}}\). These selected SBSs are marked as ASBSs (collectively represented by \(\Re_{\text{ASBSs}}\)), while the remaining SBSs are marked as OSBSs (collectively represented by \(\Re_{\text{OSBS}}\)). In the ASBS marking process, the distance between all SBSs within \(\Re_{\text{non-MSBS}}\) and all SBSs within \(\Re_{\text{MSBS}}\) is denoted as \(D_{ji}\), and the shortest distance between all SBSs within \(\Re_{\text{non-MSBS}}\) and the track is denoted as \(L_{j}\). \(D_{ji}\) and \(L_{j}\) are calculated as follows [24]:  
\[D_{ji}=\sqrt{\left(X_{j}-X_{i}\right)^{2}+\left(Y_{j}-Y_{i}\right)^{2}},j\in \Re_{\text{non-MSBS}},\ i\in\Re_{\text{MSBS}}, \tag{6}\]  
\[L_{j}=\frac{\left|mX_{j}-Y_{j}+b\right|}{\sqrt{m^{2}+1}},\ j\in\Re_{\text{non- MSBS}}. \tag{7}\]  
SBS \(j\) is marked as an ASBS when \(D_{ji}\leq\sqrt{2}\) or \(L_{j}\leq\sqrt{2}\) or as an OSBS if these criteria are not met. These criteria are visualized as a conceptual diagram in Figure 8. The rationale for establishing these criteria is as follows: because \(2R\) (i.e., super mode) is set as the signal coverage radius of an MSBS in the actual geographic coordinate system, SBS \(j\) is able support and share the load handled by SBS \(i\) or provide coverage to OIS when the distance between SBS \(i\) from \(\Re_{\text{MSBS}}\) and SBS \(j\) from \(\Re_{\text{non-MSBS}}\) is within \(2R\), or when the shortest distance between SBS \(j\) and the track is within \(2R\). Moreover, the conversion of the actual geographic coordinate system to the kernel coordinate system shows that length \(2R\) in the actual geographic coordinate system is equivalent to length \(\sqrt{2}\) in the kernel coordinate system. These ASBSs are, in principle, switched to the appropriate mode based on their respective \(\gamma_{i}\).