## 3 Model Assumption and Toss-and-Catch Algorithm
### Score-Based Selection Mechanism
#### 3.3.2 Scoring and Ranking
To the simplify the process, the Friis free space model [25] was used to calculate \(P_{\text{RX}}\); hence, RSS can be further expressed as  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]3  
where \(P_{\text{TX}}\) is the power emitted by the transmitter antenna, \(G_{\text{TX}}\) is transmitter antenna gain, \(G_{\text{RX}}\) is the receiver antenna gain, \(\lambda\) is electromagnetic wave wavelength, \(l\) is the distance between the receiver and transmitter, and \(L\) is the system dissipation factor. For the same SBS, \(\frac{G_{\text{TX}}G_{\text{RX}}\lambda^{2}}{(4\pi)^{2}l}\) is a fixed value defined as \(K\); hence, RSS can be simplified into  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]4  
To facilitate calculations, we defined RSSs as  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]5  
which serves as a substitute for calculating RSS. Let \((P_{Sx},P_{Sy})\) denote the center point of the two cut points \((P_{51x},P_{51y})\) and \((P_{52x},P_{52y})\). The maximum and minimum RSSs values of the SBS \(i\) are found at \((P_{Sx},P_{Sy})\) and \((P_{51x},P_{51y})\)(or \((P_{52x},P_{52y})\)), and the corresponding \(l\) values are, respectively, defined as \(l_{\text{max}}\) and \(l_{\text{min}}\)  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]6  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]7  
By standardizing RSSs, the score for the maximum RSS available to the OIS, namely \(score_{\text{RSSmax}}\), can be expressed as  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]8  
The larger the value of \(l\), the lower the score. A score of 0 is given when \(l=d_{i}\). When \(l<d_{i}\), the shorter the distance between the OIS and the SBS, the higher the score with 100 being the maximum. Figure 13 presents the following three scenarios: (1) the track equation is tangent to the SBS; (2) the track equation is secant to the SBS; and (3) the track equation passes the SBS center. In Scenario 1, \(l=d_{i}\) and \(score_{\text{RSSmax}}\) is therefore 0; in Scenario 2, \(l=0.4931d_{i}\) and \(score_{\text{RSSmax}}\) is therefore 87.1763; and in Scenario 3, \(l=0\), and \(score_{\text{RSSmax}}\) is therefore 100. (3) Scoring of OIS load that can be processed (\(score_{\text{load}}\))  
In addition to the two considerations discussed above, we also took into account whether the selected SBS can handle the OIS's load; hence, OIS must determine the proportion of its load that the SBS can handle. To simplify subsequent descriptions, we denoted \(\phi_{i}\) as the no-load ratio of SBS \(i\) and defined it as  
\[(P_{\text{S1}x},P_{\text{S1}y})=(\frac{-B+\sqrt{B^{2}-4 AC}}{2A},m\frac{-B+ \sqrt{B^{2}-4 AC}}{2A}+b), \tag{13}\]9  
The OIS's load ratio is denoted as \(\psi\) and obtained by substituting the load values of \"SBS \(i\)\" in the numerator and denominator of (1) with the load values of \"OIS\". We defined the score\(score_{\text{load}}\) for the OIS load that can be processed by SBS \(i\) as  
\[score_{\text{load}}= \begin{cases}100,&\phi_{i}\geq\psi\\ \frac{\phi_{i}}{\psi}\times 100,&\phi_{i}<\psi\end{cases} . \tag{29}\]When the OIS determines that \(\phi_{i}\geq\psi\), this indicates that SBS \(i\) can fully handle the OIS's load, at which point \(score_{\text{load}}\) is 100. When the OIS determines that \(\phi_{i}<\psi\), this indicates that SBS \(i\) can only handle a part of the OIS's load, at which point \(score_{\text{load}}\) is calculated based on the above formula. If SBS \(i\) is at full load, it would be considered unsuitable for a handover, at which point \(score_{\text{load}}\) is 0.