# Adaptive Energy Management in 5G Network Slicing: Requirements, Architecture, and Strategies
## 4 Energy Management Model Mathematical Formulation
### ILP Problem Formulation: OptTsNS
#### 4.2.3 Adaptive Management of Network Slices and Services  
Taking into account that the energy provisioning (mainly from renewable sources) in the mobile network is finite, the consumption demanded for service execution must be adapted to availability as defined in Requirement R5 in Section 3.1. To carry out this adaptive management, the MNO with the information on the available energy resource and customer requirements (i.e., characteristics of network slices and services) must execute the corresponding actions (management strategies) on each service (for all \(N\) services) to achieve energy-efficient consumption aimed at minimizing power consumption from non-renewable energy sources as established in Equation (8). Depending on the amount of energy available and demanded (e.g., \(P_{D}>P_{A}\)) and the features of customers, the MNO has the possibility of executing different management strategies on the services such as intra-slice priority scheme, service rejection, degradation in service quality, and use of time shifting in service execution, as indicated in Section 3.2. The execution of these strategies allows the MNO to establish an access control to the energy resources associated with the creation of the network slices and configuration of corresponding services. To find the optimal allocation of energy resources for service execution, the MNO must analyze the action of the different strategies on the services, considering the particularities of each service and/or network slice. This procedure is carried out through algorithms implemented in the core network, as shown in Figure 2b, and can be computationally very demanding, as described in detail in Section 5.2. At the end of this analysis, the MNO obtains the optimal scheduling for the set of \(N\) services (i.e., the number of services to be executed, execution times, and respective consumptions) that allows adaptive energy consumption and the minimization of energy from non-renewable sources. Through optimal scheduling of the set of \(N\) services (set defined in the proposal as _combination of services_, _comb_) the MNO tries to process as many services as possible (in the worst case the services with lower priority can be rejected if \(P_{D}>P_{A}\)) and with the minimum impact on the requirements from customers (respecting as much as possible the original execution time and power level demanded by services). In our proposed model, the adaptive energy consumption is represented by a cost function, as shown Equation (17). In this equation, \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) can be in the range [0,1] and correspond to the weights of the individual cost functions related to the management strategies, which are listed as follows:  
\[\forall i\in L,\forall j\in M,\forall k\in N:Cost_{comb}^{ij,k}=\alpha\times Cost _{AR_{comb}}+\beta\times Cost_{pri_{comb}}+\gamma\times Cost_{u_{comb}}+\delta \times Cost_{q_{comb}} \tag{17}\]  
* \(Cost_{AR_{comb}}\): Cost function related to the processing of services and defined by Equation (18). * \(Cost_{pri_{comb}}\): Cost function related to the priority level of services and defined by Equation (19). * \(Cost_{u_{comb}}\): Cost function related to the time shifting application in service execution and defined by Equation (20)
* \(Cost_{q_{comb}}\): Cost function related to the quality degradation of services and defined by Equation (21). \[Cost_{AR_{comb}}=\left\{ \begin{array}{ll}0&\text{if all services are processed,}\\ \text{Total rejected services}&\text{otherwise.}\end{array} \right. \tag{18}\]  
\[Cost_{pri_{comb}}=\left\{ \begin{array}{ll}0&\text{if all services have the maximum priority level}\\ &(j=1),\\ \sum_{i=1}^{L}\sum_{k=1}^{N}v_{k}^{i}&\text{otherwise.}\end{array} \right. \tag{19}\]  
\[Cost_{u_{comb}}=\left\{ \begin{array}{ll}0&\text{if all services are processed without time shifting}\\ &(j=1),\\ \sum_{i=1}^{L}\sum_{j=2}^{M}\sum_{k=1}^{N}u_{k,j}^{i}&\text{otherwise.}\end{array} \right. \tag{20}\]\[Cost_{q_{comb}}=\left\{