## 4 Problem Formulation and Transformation  
The optimization objective is to establish tripartite matching among LEO satellites, HAP drones, and traffic types by choosing the most suitable LEO-drone matching matrix \(\mathbf{X}_{l\times l}[m]\) and traffic-drone matching matrix \(\mathbf{Y}_{I\times Q}[m]\) in each time slot, so as to maximize the average transmitted traffic value of the network. The objective function and constraints can be formulated as follows:  
\[\underset{\mathbf{X},\mathbf{Y}}{\text{max}} \lim_{M\rightarrow\infty}\frac{U_{\text{total}}}{M}\] (10a) s.t. \[\sum_{j=1}^{J}x_{i,j}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall m \in\mathcal{M}, \tag{10b}\] \[\sum_{i=1}^{I}x_{i,j}[m]=L,\forall j\in\mathcal{J},\forall m\in \mathcal{M},\] (10c) \[\sum_{q=1}^{Q}y_{i,q}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall m \in\mathcal{M},\] (10d) \[x_{i,j}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall j\in \mathcal{J},\forall m\in\mathcal{M},\] (10e) \[y_{i,q}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall q\in \mathcal{Q},\forall m\in\mathcal{M}. \tag{10f}\]  
Constraint (10b) specifies that each HAP drone can connect to a maximum of one LEO satellite in each time slot. Constraint (10c) specifies that the number of HAP drones served by each LEO satellite is equal to the beam number \(L\). Note that even though each LEO satellite can provide services for less than \(L\) HAP drones, this would lead to inefficient use of satellite beams. Thus, in order to achieve maximum average transmitted traffic value of the network in each time slot, all beams of each satellite will be utilized. Constraint (10d) specifies that each HAP drone can transmit a maximum of one traffic type in each time slot. (10e) and (10f) are restrictions on the elements of the LEO-drone matching matrix and the traffic-drone matching matrix, respectively. The formulated problem (10a) is a mixed integer nonlinear programming problem. In the following content, we will analyze and simplify it. Given a specific \(\mathbf{X}_{l\times l}[m]\) and substituting (9) into (10a), the original problem is as follows:  
\[\underset{\mathbf{Y}}{\text{max}} \lim_{M\rightarrow\infty} \frac{1}{M}\sum_{m=1}^{M}\sum_{i=1}^{I}\left(\sum_{q=1}^{Q}u_{q} \cdot\widetilde{U}_{i,q}[m]\right)\] (11a) s.t. \[\sum_{q=1}^{Q}y_{i,q}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall m \in\mathcal{M}, \tag{11b}\] \[y_{i,q}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall q\in \mathcal{Q},\forall m\in\mathcal{M}. \tag{11c}\]  
Through analysis, it becomes evident that \(\sum_{q=1}^{Q}u_{q}\cdot\widetilde{U}_{i,q}[m]\) is solely dependent on the matching \(\{y_{i,q}[m]|q\in\mathcal{Q}\}\) between the \(i\)-th HAP drone and all traffic types in the \(m\)-th time slot, and is independent of the matching \(\left\{y_{i^{\prime},q}[m^{\prime}]|q\in\mathcal{Q},i^{\prime}\
eq i,m^{ \prime}\
eq m\right\}\) between other HAP drones and traffic types in other time slots. Consequently, maximizing (11a) can be achieved by maximizing each term within the brackets of (11a). Thus, (11a) can be rephrased as follows:  
\[\lim_{M\rightarrow\infty}\frac{1}{M}\sum_{m=1}^{M}\sum_{i=1}^{I} \left(\underset{y_{i,q}[m]|q\in\mathcal{Q}}{\max}\sum_{q=1}^{Q}\mu_{q}\cdot \widetilde{U}_{i,q}[m]\right)\] (12a) s.t. \[\sum_{j=1}^{J}x_{i,j}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall m \in\mathcal{M}, \tag{10b}\]0 \[\sum_{j=1}^{J}x_{i,j}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall m \in\mathcal{M}, \tag{10b}\]1Formulation (12a) is equivalent to optimizing \(I\times M\) independent sub-subproblems. For \(\forall i\in\mathcal{I},\forall m\in\mathcal{M}\), the sub-subproblem can be formulated as follows:  
\[\sum_{j=1}^{J}x_{i,j}[m]\in\{0,1\},\forall i\in\mathcal{I},\forall m \in\mathcal{M}, \tag{10b}\]2 (13a) s.t.