## 3 Model Assumption and Toss-and-Catch Algorithm  
The scenario being considered is presented in Figure 2, which shows a moving train with one OIS in every car. Each OIS runs the same toss-and-catch algorithm independently such that the OISs do not influence each other. Based on this assumption, this study only looks at and describes a single OIS. As for SBSs, they are assumed to be randomly deployed within the vicinity of the track. The SBSs are categorized into main SBSs (MSBSs), auxiliary SBSs (ASBSs), and other SBSs (OSBs). MSBSs are the prioritized SBSs for conducting data transmission and handovers with OISs. ASBSs support MSBSs and are used when an OIS requires a handover but is unable to identify a suitable MSBS or there are no available MSBSs within the vicinity. All other SBSs that are not MSBSs or ASBSs are categorized as OSBSs. The SBS categorization is detailed in Section 3.1.2.  
It is assumed that every SBS is equipped with the same hardware, including a GPS device with high positioning accuracy. Moreover, the OIS and every SBS are assumed to have the same capacity. The preset signal coverage radius of an activated SBS is \(R\) (measured in meter), which can be adjusted to a maximum of \(2R\) when the SBS's load increases. To establish the relationship between change in load and coverage radius, we defined \(\gamma_{i}\) as the load ratio of SBS \(i\) and expressed it as  
\[\gamma_{i}=\frac{\alpha_{i}}{\beta_{i}}. \tag{1}\]  
Figure 2: Scenario diagram.  
In (1), \(\alpha_{i}\) represents the current load of SBS \(i\) (the current load refers to the sum of the traffic being or to be processed by the SBS), \(\beta_{i}\) represents the total capacity of SBS \(i\), and \(0\leq\gamma_{i}\leq 1\). Assuming that the load is evenly distributed within the geographical environment, the coverage radius of the SBS varies alongside the load. We denoted \(d_{i}\) as the coverage radius of SBS \(i\) and defined it as  
\[d_{i}=\begin{cases}R,&0\leq\gamma_{i}\leq 0.5\\ (1+\gamma_{i})\times R,&0.5<\gamma_{i}\leq 1\end{cases}. \tag{2}\]  
Based on \(\gamma_{i}\), we then established three modes for SBS \(i\): namely, the boot preset mode, the normal mode, and the super mode. boot preset mode: when \(0\leq\gamma_{i}\leq 0.5\), \(d_{i}=R\). Normal mode: when \(0.5<\gamma_{i}<1\), \(d_{i}=(1+\gamma_{i})\times R\). Super mode: when \(\gamma_{i}=1\), \(d_{i}=2R\). In addition, the coverage radius of the OIS was also set to and fixed at \(2R\), so as to increase the likelihood of mutual data transmission and reception occurring between the SBSs deployed in the vicinity of the track.  
Figure 3 shows the flowchart of the proposed toss-and-catch algorithm. The algorithm's operation can be broken down into five stages. Stage 1: The OIS obtains information about the route ahead and the SBSs deployed around this route. This stage covers track path calculation and coordinate conversion, and SBS categorization and mode setting. Stage 2: The GPS positioning of the OIS is performed using reliable GPS equipment to obtain its real-time coordinates. Stage 3: A score-based selection mechanism is used to identify and log the most suitable SBS for the handover to be performed during the next evaluation time point. Stage 4: The overload support mechanism comes into play when the score-based mechanism fails to identify a suitable SBS. In this situation, the overload support mechanism is used to appropriately adjust the load levels of nearby SBSs or to increase the number of SBSs available to the OIS, and the updated information concerning the SBSs' sizes and modes are then used as the basis for the next round of calculations. Stage 5: During the handover stage, a handover to the most suitable SBS (as identified through the score-based selection mechanism) is performed.