## 4 Decoupled Computation Offloading and Resource Allocation with Coordinate Descent (CD)
(**a**) Default setup where the maximal local computing frequency \(Fc=1\times 10^{8}\) cycles/s. (**b**) The others are the same as the default, except for \(Fc=5\times 10^{7}\) cycles/s. According to Figure 3, the optimal offloading decision under the default settings was to process tasks \(t_{1}^{f}\), \(t_{4}^{f}\) and \(t_{9}^{f}\) locally and to offload tasks of the other seven UDs to the edge for processing. When the emergency factor (\(e_{1}^{f}\)) of a locally processed task took on a small value, nothing happened except that \(\vartheta\) changed correspondingly. We can see that, when \(e_{1}^{f}\) took on the values \(\{2^{-4},2^{-3},2^{-2},2^{-1}\}\) successively, UD 1 still processed task \(t_{1}^{f}\) locally, and the bandwidth allocation (shown in Figure 4) and edge computing resource allocation (shown in Figure 5) for UD 2, UD 3, UD 5, UD 6, UD 7, UD 8 and UD 10 remained unchanged. However, as \(e_{1}^{f}\) became large enough (\(e_{1}^{f}=2\)), UD 1 started to offload task \(t_{1}^{f}\) and was allocated some bandwidth and computing resources. Meanwhile, UD 3 and UD 10 were crowded out of resources and processed their tasks locally. As \(e_{1}^{f}\) continued to increase, more and more devices started to process their tasks locally. When \(e_{1}^{f}\) became extremely large, UD 1 monopolized all resources in the system. \[\underset{m,m,m,m,m,m,m,m}{Maximize}:H =\sum_{m\in\mathcal{M}_{1}}e_{m}^{f}(T_{m}^{f}-\frac{l_{m}^{f}}{ \alpha_{m}B_{n}\text{log}_{2}(1+\frac{p_{m}h_{m}^{2}}{\sigma^{2}})}-\frac{\rho_ {m}^{f}l_{m}^{f}}{\beta_{m}F_{e}})\]3 Computing resource allocation of UDs versus \(e_{1}^{f}\). It is noteworthy that when \(e_{1}^{f}\) shifted from one to two, not all the released bandwidth from UD 3 and UD 10 was allocated to UD 1. This can be explained by \[\underset{m,m,m,m,m,m,m,m}{Maximize}:H =\sum_{m\in\mathcal{M}_{1}}e_{m}^{f}(T_{m}^{f}-\frac{l_{m}^{f}}{ \alpha_{m}B_{n}\text{log}_{2}(1+\frac{p_{m}h_{m}^{2}}{\sigma^{2}})}-\frac{\rho_ {m}^{f}l_{m}^{f}}{\beta_{m}F_{e}})\]4 We know from Equation (21) that the bandwidth allocation share (\(\alpha_{m}\)) is proportional to \(e_{m}^{f}\). When \(e_{1}^{f}\) took the value two, the corresponding \(\vartheta_{1}\) was 1.180 (normalized), and \(\vartheta_{1}<\vartheta_{3}+\vartheta_{10}\). Therefore, the released bandwidth from UD 3 and UD 10 was reallocated to UD 1 and the remaining offloading UDs (UD 2, UD 5, UD 6, UD 7 and UD 8). It is worth noting that as \(e_{1}^{f}\) became larger, existing offloading UDs with larger \(\vartheta_{1}\) began to process tasks locally at first. However, UD 9, with the largest \(\vartheta_{9}\), could only process tasks locally all the time, while UD 1, with the smallest \(\vartheta_{1}\), could only process tasks locally at first. Fortunately, when UD 1 obtained a larger \(\vartheta_{1}\) (\(\vartheta_{1}>\vartheta_{9}\)) due to a larger \(e_{1}^{f}\), UD 1 could not only offload task \(t_{1}^{f}\) to the edge for processing but also obtain a large share of resources. This indicated that \(\binom{f}{m}\) could effectively regulate resource allocation and offloading decisions among UDs. Figure 7 shows the processing delay of each task in the system and the total revenue when \(e_{1}^{f}\) takes different values. When UD 1 began to offload its task for edge processing (the emergency factor of UD 1 took the value two), both the total delay of each task in the system and the system revenue increased. This was because a larger \(e_{1}^{f}\) indicated that the system favored UD1 in the resource allocation and received a larger reward for prioritizing UD 1. As a result, other UDs lost the opportunity to offload their tasks to the edge for processing. When \(e_{m}^{f}=256\), the completion time of all other UDs reached the maximum because their tasks were processed locally. Although the reward increased significantly when \(e_{1}^{f}\) varied from 16 to 256, the total delay of all UDs increased because the edge resources were exclusively occupied by UD 1.