# Exploiting Duplications for Efficient Task Offloading in Multi-User Edge Computing
## 3 System Model
### Communication and Computation Model  
We consider a multi-access edge network scenario, and each IoT device can select only one edge server to offload its subtask or the possible duplicated subtask. Let \(d_{i}\) be an indicator to denote where IoT device \(i\) execute its subtasks:  
\[d_{i}=\left\{ \begin{array}{ll}-i&\quad\text{if the application executed locally}\\ k&\quad\text{if it uploads the task to server $k$}\end{array} \right. \tag{1}\]  
Thus, for any subtask of user \(i\), the offloading scheme \(d_{ij}=-i\) means this subtask will be executed locally, \(d_{ij}=k\) denotes it will be executed with the help of server \(k\). Especially, if \(d_{ij^{\prime}}=d_{ij}\) represents that the subtask \(T_{i,j}\) has no duplications. We adopt \(D_{i}=\{d_{i,1},d_{i,1^{\prime}},d_{i,2},d_{i,2^{\prime}},\ldots,d_{i,|v_{i}| }\}\) to represent the a possible offloading decision for all subtasks of device \(i\), so the \(\mathbf{D}=D_{1}\times D_{2}\times\ldots\times D_{N}\) define a possible offloading strategy of all the users in the edge network. Furthermore, we can compute the wireless communication rate between user \(i\) and edge server \(k\) for a given offloading decision according to the Shannon Equation:  
\[Rate_{i,k}(\mathbf{D})=Blog_{2}(1+\frac{q_{i}\mathcal{G}_{i,k}}{\varpi+\sum_ {j\in N:D_{i}=D_{j}}q_{i}\mathcal{G}_{j,s}}) \tag{2}\]  
Here, we can see that the wireless transmission rate is affected by the bandwidth \(B\), the transmission power of the device \(q_{i}\), and most important influence factor is the offloading strategy of the nearby user: \(\sum_{j\in N:D_{i}=D_{j}}q_{j}\mathcal{G}_{j,s}\). The more users select the same edge server to execute their task. The more drastic interference will be suffered by the users. Thus, we coordinate the offloading strategies (a more reasonable \(\mathbf{D}\)) of the nearby user devices to obtain the most suitable wireless transmission rate for edge computing. We assume that the users will keep connected with the edge servers during the task execution. It should be stressed here that the Shannon Equation tells the maximum rate at which data can be transmitted over a communication channel of the specified bandwidth in the presence of noise. The noise includes not only the background noise but also the same-frequency inference. The Shannon Equation can well express the wireless transmission time of physical layer channel access scheme (e.g., CDMA) in cellular communication scenarios. We can also extend our study to some WiFi-like protocols (e.g., CSMA). In this case, the wireless transmission rate can be obtained as follows:  
\[Rate_{i,k}(\mathbf{D})=R_{n}(\frac{B_{n}}{B_{n}+\sum_{j\in N:D_{i}=D_{j}}B_{j}}) \tag{3}\]  
where \(R_{n}\) denotes the data rate of the user if it can transmit the data through the channel all its own. \(B_{n}\) denotes the weight of user n in channel sharing. In this case, the transmission rate of each user depends on the weight (\(B_{j}\)) of each user. The execution time of subtask \(T_{i,j}\) in the processor \(k\) is easily obtained as:  
\[t_{i,j,k}=c_{ij}/f_{k}\quad\forall k\in\{i\}\cup\mathcal{M} \tag{4}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(E_{i}^{c}\) & The computation consumption of user devices \\ \hline \(E_{i}^{t}\) & The transmission consumption of user devices \\ \hline \(E_{i}^{t}\) & The overall energy consumption of user devices \\ \hline \(I(i,j)\) & An indicator of duplicate subtask \\ \hline \(m_{i}(i,j)\) & The required CPU cycle to accomplish the \(T_{i,j}\) \\ \hline
**I** & A possible duplication strategy of all subtasks in the edge \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._where the \(c_{i,j}\) means the CPU cycle required to accomplish the \(T_{i,j}\), and the \(f_{k}\) denotes the computation capability of user \(i\) or the edge server \(k\). In our scenario, the actual finish time of subtask \(T_{i,j}\) in server \(k\) is more complex, as it is affected by three factors: (1) the execution time in server \(k\).