## 3 System Model  
As shown in Figure 1, the system works in an OFDMA manner and consists of a base station serving \(M\) UDs from a set \(\mathcal{M}=\{1,2,3,\cdots,M\}\). The BS is endowed with a bandwidth \(B\) (in hertz) and connected with an edge server whose computing capacity is denoted as \(Fe\) (in CPU cycles). AP, BS and the edge are used interchangeably in the remainder of this article. Multiple UDs undertaking \(F\) types of computation tasks contend for resources to shorten task completion time. In this paper, we classified computation tasks into four categories (i.e., \(F=4\)) based on their data size, i.e., tasks of small size, tasks of regular size, tasks of large size and tasks of immense size.  
A computation task is denoted as a quadruplet \(o_{m}=(\rho_{m}^{f},l_{m}^{f},e_{m}^{f},\,T_{m}^{f})\). \(\rho_{m}^{f}\) denotes the number of CPU cycles required to process one bit of task \(t_{m}^{f}\) (\(f\in\{1,2,\cdots,f,\tau\}\)) on UD \(m\). \(l_{m}^{f}\) (in bits) represents the data size of task \(t_{m}^{f}\). \(e_{m}^{f}\) is the emergency factor of task \(t_{m}^{f}\), which can be utilized to regulate resource allocation and offloading decisions. \(T_{m}^{f}\) indicates the maximum acceptable processing delay of \(t_{m}^{f}\). It is worth mentioning that although the emergency factor is described as an inherent part of the task, it can also be defined as a configurable parameter managed by the BS. \(\boldsymbol{o}=\{o_{m}\}^{M}\) contains all the task information from UDs requesting a task offloading. Instead of the arbitrary divisible task processing model, the binary task processing model was considered in this paper. It was assumed that a task was either completed locally (\(x_{m}=0\)) or at the AP (\(x_{m}=1\)), where \(x_{m}\) is the task offloading decision of UD \(m\). Once \(x_{m}=1\), the AP has to allocate \(\a  
Figure 1: System model.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Work** & **Offloading Mode** & **Optimization Variables** & **Objective** & **Methodology** \\ \hline
[30] & Binary & \(\boldsymbol{x}\), \(\boldsymbol{b}\), \(\boldsymbol{\alpha}\) & R [6] & Lyapunov optimization and DRL \\ \hline Our work & Binary & \(\boldsymbol{x}\), \(\boldsymbol{b}\), \(\boldsymbol{\alpha}\), \(\boldsymbol{p}\) & Revenue maximization & CD and Lagrange multiplier method \\ \hline \end{tabular}  
* \(\boldsymbol{x}\) denotes the offloading decision vector; 2: \(\boldsymbol{b}\) denotes the communication resource allocation vector; 3: \(\boldsymbol{\alpha}\) denotes the edge-computing resource allocation vector; 4: D stands for latency/delay minimization; 5: E stands for energy consumption minimization; 6: R stands for computation rate maximization; 7: denotes the computation node selection vector; 8: A denotes the splitting ratio; 9: \(\boldsymbol{p}\) denotes the transmission power.  
\end{table}
Table 1: \(Cont.\)wireless bandwidth and \(\beta_{m}\in(0,1]\) of its computing resources to task \(t_{m}^{f}\). All the resources allocated to UDs should not exceed the AP's capacity,  
\[\sum_{m=1}^{M}x_{m}\alpha_{m}\leq 1, \tag{1}\]  
\[\sum_{m=1}^{M}x_{m}\beta_{m}\leq 1. \tag{2}\]  
In this study, we focused on the offloading decision and resource allocation within a scheduling slot. It was assumed that each user device (UD) had at most one task to process and the channel status between each UD and the base station was assumed to be quasi-static.