## 3 Problem Formulation  
In this paper, an optimization problem is formulated to jointly optimize the UAV's trajectory, UAV's available battery power, vehicles' packet transmission scheduling, and the queue length at the UAV. The vehicles compute the local stochastic gradient and communicate with the UAV to seek a global optimum. Subsequently, we minimize the queue length at the UAV by reducing the transmission rate from multiple vehicles [83]. The data distribution experiences unpredictable changes and needs to periodically update the training set. The queue at the UAV follows the standard \(M/M/k\) model and is a bin set of capacity \(c_{q}\), comprising a packet set \(a_{\Psi}\), where the size of each packet is \(\Psi_{i}\). To optimize the queue length at the UAV, we formulate a bin-packing problem (BPP) as an MDP to pack all the packets into a queue such that the total number of queues is minimized for queue capacity \(c_{q}\). The variable \(\zeta_{ik}\) denotes whether packet \(i\) is assigned to queue \(k\), and the variable \(\phi_{k}\) denotes whether bin \(k\) is utilized. The problem (\(P1\)) is formulated as follows in Equation (6):  
\[ \begin{split}\mathbf{P}1:&\min_{\mathbf{\psi}^{(t)},\mathbf{ \mathcal{D}}^{(t)}}\left\{w_{1}\left[\underbrace{\sum_{t=1}^{T}\min\left(\max _{v\in\mathcal{V}}(d_{v}^{t}),\tau_{d}\right)+T\tau_{u}}_{SP_{1}}\right]+\right. \\ &\left.w_{2}\underbrace{\left[\sum_{t=1}^{T}\sum_{v\in\mathcal{V }}\mathcal{D}_{v}^{(t)}\min(d_{v}^{(t)},\tau_{d})\right]}_{SP_{2}}+w_{3} \underbrace{\left[\sum_{t=1}^{T}\sum_{v\in\mathcal{V}}\left(\sum_{j\in\mathcal{ V}}\sum_{k\in\mathcal{K}}t_{i}p_{ij}^{(k)}\xi_{i,j}^{(k)}(t)\right)\right] \right\}_{SP_{3}}\\ \end{split}  \tag{6}\]  
subject to  
\[C1: \min\sum_{v\in\mathcal{V}}\phi_{k},\phi_{k}\in\{0,1\},\forall v\in \mathcal{V} \tag{7}\] \[C2: \sum_{i\in\mathfrak{w}}\zeta_{ik}\Psi_{i}\leq c_{q\phi_{k}},\quad \forall k\in K,\forall v\in\mathcal{V}\] (8) \[C3: \sum_{k\in K}\zeta_{ik}=1\quad\forall i\in a_{\Psi},\zeta_{ik} \in\{0,1\},\forall i\in a_{\Psi}\forall k\in K\] (9) \[C4: ||\Delta_{i(t+1)}-\Delta_{i(t)}||\leq v_{max}(t)t_{i,i+1}\] (10) \[C5: ||\Delta_{i(t+1)}-\Delta_{i(t)}||\geq d_{min}\] (11) \[C6: \Delta_{(x,y)}=\Delta_{(x_{0},q_{0})}\cdot e^{-\alpha\left(\frac{ p^{2}}{x^{2}}+\frac{p^{2}}{y^{2}}\right)}+\mathcal{H}\] (12) \[C7: w_{1}+w_{2}+w_{3}=1 \tag{13}\]  
where problem (\(P1\)) is a multi-objective and multi-variable non-convex optimization problem. In problem (\(P1\)) in Equation (6), the parameter \(\mathbf{\psi}^{(t)}\) is a vector consisting of different packets comprising PFL model hyper-parameters. Rather than transmitting the local model hyper-parameters from all the vehicles to the UAV, PFL aims to select a subset of vehicles \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(\tilde{\tau}_{T}\) & Global model, collection of sequential data gathered during repeated transmissions \\ \(p_{\theta}\) & Probability of state transitions of UAV \\ \(\theta\) & Set of parameters of local models \\ \(A_{t}\) & Attention mechanism to capture dominant past states’ influence on future state transitions \\ \(m_{\phi}(\vec{\mathbf{\Upsilon}}_{T}\,|\,\tilde{\mathbf{\Upsilon}}_{T})\) & Variational distribution that approximates the posterior \(p_{\theta}(\vec{\mathbf{\Upsilon}}_{T}\,|\,\tilde{\mathbf{\Upsilon}}_{T})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._that communicate their local model hyper-parameters to the UAV in a TTI. The variable \(\tau_{d}\) is the delay experienced by a vehicle while downloading the global model hyper-parameters from the UAV. The term \(\tau_{u}\) implies the delay experienced by a vehicle while uploading its local model hyper-parameters to the UAV. The variable \(t_{i}\) is the time duration of the TTI under consideration and \(p_{i,j}\) is the power consumption of a UAV in a TTI. The constraint \(C1\) emphasizes the decision variable \(\phi_{k}\) that denotes whether bin \(k\) is utilized in a transmission window or is idle.