## 7 IoT Wireless Resource Allocation Scheme  
When clients are homogeneous, it is sufficient to reduce the participation of low-contribution clients in aggregation, thereby allocating more bandwidth resources to high-contribution clients. This allows for more local training to reduce the edge's local loss function, thereby influencing the global average loss function. However, when clients are heterogeneous, it is necessary to consider that high-contribution clients may consume significant energy and take longer computation times during local calculations. This means they cannot perform multiple iterations within the same energy consumption or time frame. Therefore, this section comprehensively considers contribution and heterogeneous computing capability, selecting clients and allocating bandwidth resources based on the principle of fairness. The main idea is to optimize client selection within a given time frame during the asynchronous aggregation phase. Specifically, this involves adjusting whether each client \(i\) participates in the asynchronous aggregation during the \(r\)th round of global aggregation, denoted as \(\alpha_{r,i}^{a}\). From a fairness perspective, the contribution of edges and clients (obtained by calculating the Shapley Value (SV) or model similarity) within the same number of training and aggregation cycles can directly reflect data quality [20]. According to the approach from the previous section, edges or clients with higher computing capability should inherently possess stronger data feature extraction capabilities, while those with lower computing capability should be given some tolerance in terms of model performance. In the specific algorithm, this is reflected as follows: when it is necessary to exclude an edge with a low SV, the maximum computing resource \(f_{j}^{max}\) of the clients under that edge must be considered (since the number of local training iterations after dropout depends on the maximum computing capability under the edge). If \(f_{j}^{max}\) ranks low among all computing capabilitys, it is considered tolerable and is not excluded; however, if \(f_{j}^{max}\) ranks high, it is considered intolerable and can be excluded. Similarly, when using model similarity to assess client contribution and selecting the top clients, the computing capability \(f_{i}\) of clients under each edge needs to be sorted from high to low. If a client has low contribution and ranks low in computing capability, it is considered tolerable, as its data have not been reasonably utilized due to limited computing resources. By allocating more bandwidth resources and training opportunities to it, its model contribution can be improved. Conversely, if a client has low contribution but ranks high in computing capability, it indicates that its local data quality is low. Even with high computing capability, its contribution to the global model is minimal and it is considered intolerable, thus excluded from the client selection strategy. Algorithm 3 provides a detailed explanation using the rapid convergence phase as an example, with the same logic applying to the performance enhancement phase. In each round, client computation (local updates) for selected participants is \(\mathcal{O}(K_{s}\tau_{1}d)\), where \(K_{s}\) is the number of participating clients. Contribution ranking and participation decision per edge costs \(\mathcal{O}(K\log K)\), assuming sorted ranking. Bandwidth reallocation and edge-level aggregation per edge is \(\mathcal{O}(K_{j}d)\), with \(K_{j}\) clients under edge \(j\). Thus, the overall complexity per round is \(\mathcal{O}\big{(}K_{s}\tau_{1}d+NK\log K+NK_{j}d\big{)}\), which scales efficiently due to selective update and avoids full participation overhead. Additionally, the ranking mechanism filters out high-delay or low-impact clients, reducing unnecessary gradient computation and improving resource utilization.