## 4 Decoupled Computation Offloading and Resource Allocation with Coordinate Descent (CD)
Our scheme penalized tasks of immense size by setting their emergency factor to \(\frac{\overline{L^{\pi}}}{\varepsilon_{p}\overline{L^{\pi}}}\), where \(\overline{L^{\pi}}\) is the average of regular size, \(\epsilon_{p}\) is the penalty coefficient and \(L^{im}\) is the data volume of the task with immense data size. Conversely, we supported tasks of small size by setting their emergency factor to \(\frac{\epsilon_{p}\overline{L^{\pi}}}{\overline{L^{\pi}}}\), where \(\epsilon_{e}\) is the enhancement coefficient and \(L^{sm}\) is the data volume of the task with small data size. The baseline schemes used in this paper included:

* All offload (AO): all tasks are processed at the edge server. * All local (AL): all tasks are processed locally. * Random offload (RO): the offloading decision is randomly generated and the resource allocation decisions are obtained with Algorithm 2. * Brute-force search method (BF): searches all the \(2^{M}\) offloading schemes and selects the one with the highest reward as the final solution. * Naive coordinate descent (NCD): directly goes into the \"while loop\" [16] with the randomly initialized \(x^{0}\) in Algorithm 1. * Deep-reinforcement-learning-based scheme (DRL): uses channel conditions and task data size to make offloading decisions and utilizes the critic module to get the resource allocation scheme with minimum delay, which is slightly different from [29]. ### Simulation Setting

By default, there were \(M=10\) UDs in our system. The channel gain of the large-scale fading model in this paper was \(A_{d}(\frac{3\times 10^{8}}{4\pi f_{0}d_{m}})^{d_{0}}\chi\)[30], where \(A_{d}\) denotes the antenna gain of a UD, \(f_{0}\) represents the carrier frequency, \(d_{m}\) is the distance between UD \(m\) and BS in meters, the path loss exponent was \(d_{0}=2.6\) and \(\chi\) followed a Rayleigh distribution with unit variance. The BS had a bandwidth of \(B=2\) MHz and a computing capacity of \(Fe=1\times 10^{10}\) cycles/second by default. The maximal transmission power was \(P_{m}^{max}=0.2\) (in watts). UD's local computing capacity \(Fc\) took the value \(Fc_{m}=1\times 10^{8}\)\(\forall m\in\mathcal{M}\) cycles/second to \(1\times 10^{9}\) cycles/second. The value of \(e^{f}\) was set to \(1\), and \(T^{f}=1\) as the maximal acceptable service delay. We considered tasks of \(F=4\) categories, and \(\rho^{f}\) was randomly taken from \(\{100,1000\}\) (in cycles/bit). ### Result Discussion

In Figure 2, we varied \(\mathbf{Fc}\) from \(Fc_{m}=1\times 10^{8}\)\(\forall m\in\mathcal{M}\) to \(Fc_{m}=5\times 10^{7}\)\(\forall m\in\mathcal{M}\). This caused tasks processed locally, as shown in Figure 2b, to time out. The results in Figure 2a demonstrate that our proposed LCD algorithm could effectively converge to the optimal scheme (results from BF) and the NCD method deviated slightly from the optimal solution. Furthermore, the resources deployed at the edge could support the simultaneous task offloading for six to eight UDs (with a data size of one megabit). When UDs exceeded that threshold, the overall revenue of the system significantly declined. However, in Figure 2a, the overall rewards remained unchanged and even slightly increased as the number of UDs increased. This was because the local computing resources were sufficient to process the tasks locally without incurring negative rewards. The system could even enhance revenue by offloading tasks from UDs with more competitive conditions (e.g., better channel conditions). This was no longer the case in Figure 2b for \(Fc_{m}=5\times 10^{7}\)\(\forall m\in\mathcal{M}\), where the revenue declined as the number of UDs increased. In this scenario, processing tasks locally resulted in negative rewards due to the timeout caused by an inadequate local processing capacity. \[\underset{m,m,m,m,m,m,m,m}{Maximize}:H =\sum_{m\in\mathcal{M}_{1}}e_{m}^{f}(T_{m}^{f}-\frac{l_{m}^{f}}{ \alpha_{m}B_{n}\text{log}_{2}(1+\frac{p_{m}h_{m}^{2}}{\sigma^{2}})}-\frac{\rho_ {m}^{f}l_{m}^{f}}{\beta_{m}F_{e}})\]2 Rewards versus UDs in the system.