## 2 System Model and Problem Formulation  
### System Model  
We investigate the issue of handover selection for numerous TTGs in satellite\(-\)to\(-\)ground communication, as illustrated in Figure 1. As depicted, the system comprises \(K\) static TTGs on the ground, each requiring continuous communication services. Above them, a constellation of LEO satellites continuously orbits. For each TTG, there is always one'serving satellite' providing active communication, while several other satellites are visible and available for a potential handover, forming a dynamic set of choices. All satellites are assumed to operate at the same altitude, denoted as \(h_{s}\), and maintain a constant speed \(v_{s}\). Each TTG is simultaneously covered by multiple satellites at any given time. However, due to the relative motion between the satellites and the TTGs, the visibility time of each satellite to a specific TTG is limited. When a satellite can no longer cover TTG \(k\) due to its movement, the satellite network broadcasts system information to configure a set of visible satellites for the TTG, enabling a seamless handover and sustained communication. For TTG \(k\), the number of visible satellites at time slot \(t\) is denoted as \(M_{k}^{t}\), with each satellite indexed by \(m\in\mathcal{M}_{k}^{t}=\{1,2,\cdots,M_{k}^{t}\}\). The service status for TTG \(k\) at time slot \(t\) is represented by the vector \(\mathbf{s}_{k}^{t}=(s_{k,1}^{t},s_{k,2}^{t},\cdots,s_{k,M_{k}^{t}}^{t})\), where \(s_{k,m}^{t}\) is a binary variable indicating whether satellite \(m\) is serving TTG \(k\) at time slot \(t\), defined as  
\[s_{k,m}^{t}= \begin{cases}1&\text{if TTG $k$ is currently served by satellite $m$}\\ 0&\text{otherwise.}\end{cases}  \tag{1}\]  
Note that different TTGs may have distinct resource requirements, such as throughput and latency. To effectively model these varying demands, we introduce a normalized resource demand weight \(w_{k}^{t}\) for each TTG, ranging from 0 to 1. The sum of the weights of all TTGs accessing a particular satellite indicates the satellite's load state. We assume each TTG periodically retrieves load information for its visible satellites from the System Information Block (SIB) broadcast. The load of the \(M_{k}^{t}\) visible satellites for TTG \(k\) is represented as  
\[\mathbf{I}^{t}=\{l_{1}^{t},l_{2}^{t},\cdots,l_{M_{k}^{t}}^{t}\}, \tag{2}\]  
Figure 1: An illustration of LEO satellite\(-\)to\(-\)ground communication system with handover selection for multiple TTGs. where \(l_{m}^{t}\) denotes the sum of the normalized demand weights of all the TTGs accessing satellite \(m\). We assume that TTG \(k\) accurately determines its location via Global Navigation Satellite System (GNSS) services, denoted as \(\mathbf{p}_{k}=[\xi_{k},y_{k},0]\in\mathbb{R}_{+}^{3}\), where \(\xi_{k}\) and \(y_{k}\) denote its fixed horizontal coordinates. Additionally, TTG \(k\) can obtain the position and velocity of its visible satellites at time slot \(t\) from the broadcast ephemeris [18], denoted as \(\mathbf{q}_{m}^{t}=[x_{m}^{t},y_{m}^{t},h_{s}]\in\mathbb{R}_{+}^{3}\), where \(m\in\mathcal{M}_{k}^{t}\), and \(x_{m}^{t}\) and \(y_{m}^{t}\) are the satellite's time\(-\)varying horizontal coordinates.