# Energy-Efficient Resource Provisioning Strategy for Reduced Power Consumption in Edge Computing +
## 3 Model and Problem Formulation
### Problem Formulations  
In this subsection, we consider minimizing the total energy consumption of users under the latency constraints. Based on this hierarchical structure in Figure 1, users can process the requests either at local devices or remote edge nodes and return the processing results to the user. For user \(u_{i}\), we use \(E_{c}(u_{i})\) to denote the power consumption for executing task \(i\) at local device as shown in Equation (1). \(f_{i}\) is a superlinear function denoting the CPU energy consumption of device \(i\). \(\alpha\) is a pre-configured model parameter depending on the chip architecture [28; 29; 30; 31].  
\[E_{c}(u_{i})=\alpha\cdot f_{i}^{3} \tag{1}\]  
We use \(E_{t}(u_{i})\) to denote the energy consumption for executing task \(i\) at remote edge node as shown in Equation (2).  
Figure 2: Probability of the emit interval in different configurations.  
\[E_{t}(u_{i})=p_{i}\cdot D_{t}=\frac{p_{i}\cdot T_{i}}{r_{ij}} \tag{2}\]  
Let \(p_{i}\) denote the transmit power of device \(u_{i}\). Let \(D_{i}^{t}\) denote the transmission delay which decides by the size of task \(T_{i}\) and the transmission speed \(r_{ij}\). The channel power gain from the device to the edge node can be represented by \(H(d_{ij})=hg_{0}(d_{0}/d_{ij})^{\theta}\), where \(g_{0}\) is the path-loss constant, \(d_{0}\) is the reference distance, \(\theta\) is the path-loss exponent, and \(d_{ij}\) is the distance from device \(u_{i}\) to sensor \(v_{j}\) that it connects. Let \(N_{0}\) denote the channel white Guassian noise, and \(\omega\) denote the channel path loss exponent. Hence, the speed of offloaded tasks at edge node \(v_{j}\) is shown in Equation (3).  
\[r_{ij}=B\cdot log_{2}(1+\frac{H(d_{ij})\cdot p_{i}}{N_{0}\cdot\omega}) \tag{3}\]  
Let \(E(u_{i})\) denote the total energy consumption of device \(u_{i}\), where \(E(u_{i})=E_{t}(u_{i})+E_{c}(u_{i})\). In this paper, we formulate the resource provision problem for multiple users in edge computing. Our objective is to find an appropriate scheme for the set of users \(U\) that minimizes the total energy consumption and satisfies the constraints on the resources and the maximum tolerant delays. For each device, we use \(D_{i}^{c}\) to denote the computation delay, where \(D_{i}^{c}=M_{i}/C_{i}\). Let \(D_{i}^{t}\) denote the transmission delay, where \(D_{i}^{t}=T_{i}/r_{ij}\). The computation delay on edge node for task \(i\) is denoted as \(D_{i}^{c}\), where \(D_{i}^{c}=d_{r}(u_{i})+d_{w}(u_{i})\). Let \(d_{r}(u_{i})\) and \(d_{w}(u_{i})\) denote the processing time and waiting time of \(u_{i}\) on edge node \(v_{j}\), respectively.  
\[\text{minimize }\sum_{v_{j}\in V}\sum_{u_{j}^{c}\in U_{v_{j}}}E(u_{i}) \tag{4}\] \[\text{subject to }E(u_{i})=E_{t}(u_{i})+E_{c}(u_{i})\] (5) \[D_{i}^{c}+D_{i}^{t}+D_{i}^{c}\leq D_{i}^{d}\] (6) \[f_{i}\leq f_{i}^{max},p_{i}\leq p_{i}^{max} \tag{7}\]  
Equations (4) and (5) show the objective of minimizing the total energy consumption for the users. Equation (6) is the constraint on the deadline of the users, which means the total delay can not over the deadline \(D_{i}^{d}\) for user \(i\). Equation (7) is the constraint on the computation capacity and the transmit power of each user.