## 3 Congestion-Aware EFL-Based Model Aggregation Policy in Edge Computing
### Primary Objectives and Algorithm Designs  
A reliable model aggregation policy with MLSTM-CEFL offers four primary objectives in industrial EFL-based applications, which include minimizing model aggregation latencies, preventing excessive model drops, ensuring efficient utilization of model computing resources, and maximizing model communication throughput. In multi-service industrial \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Conditions** & **Upper-Bound Delays** & **Criticalities** & **FL-Based Scenarios** \\ \hline Class-1 & 50 ms & High mission-critical & Visual-assisted robot, low-latency manufacturing control \\ Class-2 & 150 ms & Mission-critical & Industrial visual/surveillance systems \\ Class-3 & 300 ms & Low mission-critical & Imaging compression for storage and retrieval \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The class characteristics, criticalities, and FL-based scenarios in industrial services. applications, MLSTM-CEFL distinguishes between each application class through adaptive resource pooling modifications. Consequently, the scheduling policy and offloading decisions are formulated based on these four primary outputs. To minimize the overall latencies, the three-tuple elements are jointly considered, including the latencies of local model computation, model communication, and edge aggregation. The local computing time is considered based on the size of input data batch \(D_{n(t)}\) to complete the \(M_{n(t)}\). The final local learning model at \(t\)-index is the optimal model parameter that minimizes the loss of each mini-batch input. Once the optimal loss-minimized \(M_{n(t)}^{*}\) is acquired, the updating process in a time slot \(ts\), denoted as \(T_{update}\), is executed across all IIoT participant nodes. The models are directed towards a selected edge aggregator. This process is formulated at the \(t\)-index by summing up the model updating latencies of each local node. Upon receiving the local models for the \(t\)-index, the edge aggregation procedure is performed to minimize the edge loss function within each matched industrial imaging service. Given the distinct requirements of each service for model construction and target features, traditional federated averaging algorithms are employed in this design to aggregate service \(m\) iteratively, culminating in the final learning model denoted as \(T\Big{[}M_{m}^{G(e)}\Big{]}\). Equations (1)-(3) are presented as follows: (1) the consumed latencies of local model computing from a single participant, (2) model communication between a set of matched participants-edge in a single time slot, and (3) edge computing for the aggregated model in a single time slot. The final objective is provided in Equation (4). \(T_{(n)}^{comp}\), \(T_{(e)}^{comm}\), \(T_{(e)}^{comp}\), and \(T_{update}\) denote the following descriptions: (1) computation time in the local IIoT participant \(n\), (2) communication time between a set of participants and a single edge aggregator \(e\), (3) computation time in edge aggregator \(e\), including global loss minimization and federated averaging, and (4) uplink data rate for model updating under specific IIoT network conditions. Additionally, \(L\) represents the loss function of imaging or visual-assisted model training in industrial applications capable of utilizing the mean squared error to calculate gradients within each specified convolutional neural network model. \[T_{(n)}^{comp}\Big{[}M_{n(t)}^{*}\Big{]}\triangleq T[\underset{M_{n(t)}}{ \text{argmin}}L\left(M_{n(t)}\left|D_{n(t)}\right)\right] \tag{1}\]  
\[T_{(e)}^{comm}\triangleq\sum_{ts}\sum_{n}T_{update}[M_{n(t)}^{*}] \tag{2}\]  
\[T_{(e)}^{comp}\triangleq T[L(M_{c(t)}\Big{|}\sum_{n=1}^{N}D_{n(t)})]+T[M_{m}^ {G(e)}] \tag{3}\]  
\[min\Big{(}\sum T_{(n)}^{comp}+T_{(e)}^{comm}+T_{(e)}^{comp}\Big{)}_{m}\leq\tau_ {m} \tag{4}\]  
High model drops significantly degrade the final learning accuracy within EFL-based applications, primarily due to the absence of high-quality data contributions and incomplete training procedures.