## 4 Decoupled Computation Offloading and Resource Allocation with Coordinate Descent (CD)
Similarly, when tasks of small size (tasks from UD 2 and UD 8) emerged (\(p=0.8\)), our scheme failed to obtain the minimum delay as well. However, our scheme succeeded in excluding tasks of immense size from a monopoly on edge resources and supporting tasks of small size to contend for edge resources. For example, tasks from UD 2 and UD 8 with \(p=0.8\) and tasks from UD2, UD5 and UD 10 with \(p=0.9\) obtained shorter delays when compared with the DRL scheme. ## 6 Conclusions

Current task-offloading schemes targeting a minimum delay tend to prioritize tasks of large data size, which prevents tasks of small data size from being offloaded. When coexisting with tasks of large data size, tasks of small data size may lose opportunities to be offloaded to the edge for processing. In this paper, we introduced the emergency factor to penalize tasks of immense size for monopolizing system resources and support tasks of small size in contending for system resources. The joint task offloading and resource allocation issue was formulated as an MINLP problem that aimed to maximize the processing time reward. A bisection-search-based resource allocation algorithm combined with a CD-based method was proposed to solve the problem. Simulation results validated the effectiveness of our proposed scheme in regulating offloading decision and resource allocation when there was a significant difference in the data size of the offloaded tasks. In future work, we will study resource allocation based on a more fine-grained task classification scheme and explore the use of state-of-the-art deep reinforcement learning methods [29; 33] for efficiency. We may also consider schemes for different objectives, such as profit [28] and QoS, and may also consider deploying caching [24; 31] at the edge. \[\underset{m,m,m,m,m,m,m,m}{Maximize}:H =\sum_{m\in\mathcal{M}_{1}}e_{m}^{f}(T_{m}^{f}-\frac{l_{m}^{f}}{ \alpha_{m}B_{n}\text{log}_{2}(1+\frac{p_{m}h_{m}^{2}}{\sigma^{2}})}-\frac{\rho_ {m}^{f}l_{m}^{f}}{\beta_{m}F_{e}})\]8 Delay versus the sampling probability. **Author Contributions:** Conceptualization, L.D. and H.Y. ; methodology, H.Y. ; software, L.D. ; validation, L.D. and W.H. ; formal analysis, L.D. ; investigation, L.D. ; resources, H.Y. ; data curation, L.D. and W.H. ; writing--original draft preparation, L.D. ; writing--review and editing, L.D. and W.H. ; visualization, L.D. ; supervision, H.Y. ; project administration, H.Y. ; funding acquisition, H.Y. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix A

**Proof. ** The first terms in stationarity Equations (10)-(12) are positive, which result in the positiveness of Lagrangian multipliers \(\lambda_{1},\lambda_{2},\omega\). Furthermore, the complementary slackness Equations (13)-(15) hold only when

\[\underset{m,m,m,m,m,m,m,m}{Maximize}:H =\sum_{m\in\mathcal{M}_{1}}e_{m}^{f}(T_{m}^{f}-\frac{l_{m}^{f}}{ \alpha_{m}B_{n}\text{log}_{2}(1+\frac{p_{m}h_{m}^{2}}{\sigma^{2}})}-\frac{\rho_ {m}^{f}l_{m}^{f}}{\beta_{m}F_{e}})\]9 (11) \[\text{C1}:\quad\sum_{m\in\mathcal{M}_{1}}\alpha_{m}\leq 1\]0 (12) \[p_{m}^{max}-\mathbf{p}_



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