## 2 System Model
### Cost Function  
The conventional optimization techniques and cost function with constraints have high computational overhead and algorithm complexity [43]. Therefore, we need to formu \begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Meaning** \\ \hline \(E_{di}\) & Energy consumption due to division process per component \\ \hline \(E_{li}\) & Total energy consumption to execute \(c_{i}\) locally \\ \hline \(E_{ai}\) & Total remote energy consumption for \(c_{i}\) \\ \hline \(E_{ii}\) & Transmission energy consumption for \(c_{i}\) \\ \hline \(E_{ri}\) & Reception energy consumption for \(c_{i}\) \\ \hline \(\epsilon\) & Average switch capacitance and activity factor \\ \hline \(e_{i}\) & Binary offloading decision variable \\ \hline \(\eta\) & Number of CPU cycles per bit \\ \hline \(f_{cl}(c_{i})\) & Local cost for component \(c_{i}\) \\ \hline \(f_{co}(c_{i})\) & Remote cost for component \(c_{i}\) \\ \hline \(f_{s}\) & CPU frequency of MES \\ \hline \(f_{ui}\) & CPU frequency of UE \\ \hline \(\delta_{1},\delta_{2}\) & Weighting coefficients for local cost function \\ \hline \(\delta_{3},\delta_{4}\) & Weighting coefficients for remote cost function \\ \hline \(\gamma\) & Division resolution in partitioning \\ \hline \(h_{dl_{i}},h_{up}\) & Channel fading coefficients for downlink, uplink \\ \hline \(K\) & Maximum available subcarriers \\ \hline \(k_{i}\) & Number of subcarriers assigned to \(c_{i}\) \\ \hline \(\lambda_{i}\) & Distance between UE and MES \\ \hline \(m\) & Task size \\ \hline \(\mu_{i}\) & Input data size of \(c_{i}\) \\ \hline \(N_{o}\) & Noise power \\ \hline \(n\) & Number of components per tasks \\ \hline \(\mathbf{OP}\) & Matrix of possible offloading policies \\ \hline \(op^{*}\) & Optimal partitioning \\ \hline \(\mathbf{PR}\) & Matrix of possible partitions \\ \hline \(p_{si}\) & Transmitting power of MES \\ \hline \(p_{ui}^{t}\) & Transmitting power of UE \\ \hline \(p_{ui}^{t}\) & Receiving power of UE \\ \hline \(R\) & Maximum CPU cores of MES \\ \hline \(\theta\) & Path loss exponent \\ \hline \(\tau\) & Required delay to divide a task into two components \\ \hline \(\tau_{i}\) & Number of CPU cores of MES assigned to \(c_{i}\) \\ \hline \(\rho_{i}\) & Output data size of \(c_{i}\) \\ \hline \(v_{dil}\) & Downlink data rate \\ \hline \(v_{upi}\) & Uplink data rate \\ \hline \(z^{*}\) & Optimal offloading policy \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._late a comprehensive cost function, considering all important parameters, for generating a training dataset only. The algorithm complexity for generating a training dataset through such a comprehensive cost function is also high however the computation during training phase only occurs once. After the training phase the trained DNN has constant complexity \(O(1)\)[41]. Our comprehensive cost function depends on delays and energy consumptions due to execution, transmission, reception, and task-division for partial offloading. Furthermore, the proposed cost function also considers the propagation delay, radio resources, and computing resources. In the partitioning process, if the number of components increases, the time delay and energy consumption due to task-division-process also increase. Therefore, the time delay, \(d_{di}\), due to task-division per component can be written as:  
\[d_{di}\triangleq f_{1}(n)=\frac{(n-1)\tau}{n}, \tag{13}\]  
where \(\tau\) is the time in which the UE can divide a task in two components. Similarly, the energy consumption, \(E_{di}\), per component due to partitioning of the task can be written as:  
\[E_{di}=d_{di}ef_{ui}^{3}. \tag{14}\]  
We can write the equation for the total delay, \(d_{i}\), for \(c_{i}\) as:  
\[d_{i}=\left\{ \begin{array}{ll}d_{li}+d_{di},&e_{i}=0,\\ d_{oi}+d_{di},&e_{i}=1.\end{array} \right. \tag{15}\]  
Similarly, the total energy consumption, \(E_{i}\), due to \(c_{i}\) can be written as:  
\[E_{i}=\left\{ \begin{array}{ll}E_{li}+E_{di},&e_{i}=0,\\ E_{oi}+E_{di},&e_{i}=1.\end{array} \right.