## 3 System Model and Problem Formulation  
In this section we describe the MEC system model and formulate the MECS status determination problem. Before we proceed, we summarize the notations used in this paper in Table 1. The system model considered in this paper is depicted in Figure 1. We aim to devise a discrete time controller operating periodically at the beginning of each time slot in a MEC system composed of \(M\) MECSs. Specifically, our goal is to design a controller that periodically determines the activation status of each MECS during a time slot \(t\) at the beginning of the time slot in order to optimize the energy consumption of the MEC system by taking into account the maximum service delay that the MEC system can provide during the time slot. We assume that the controller operates on a large time scale, which means that the duration of a time slot denoted by \(\tau\) is much longer than the task inter-arrival time. We denote the MECS set in the system by \(\mathcal{M}\). Each MECS \(i\) has a processing capacity \(C_{i}\) in terms of the number of CPU cycles available per time slot. Each MECS \(i\in\mathcal{M}\) has a task queue to accommodate the tasks to be processed. We assume that the task queue is served in a first-in, first-out (FIFO) manner. The workload, represented as the number of CPU cycles required by the tasks in the task queue of MECS \(i\) at the start of time slot \(t\), is denoted by \(Q_{i}(t)\). To specify the decision made by the controller at the start of each time slot \(t\), we introduce a MECS status vector that represents the activation status of each MECS during a time slot \(t\) as follows. \[\eta(t)=\{\eta_{i}(t):i\in\mathcal{M}\}, \tag{1}\]  
where \(\eta_{i}(t)\) is a binary indicator representing the operation mode of MECS \(i\) during a time slot \(t\). When \(\eta_{i}(t)=1\), \(i\) is in an active state. In this case, \(i\) can accommodate and process the tasks that have been assigned to it. On the other hand, if \(\eta_{i}(t)=0\), MECS \(i\) is in a sleep state during the time slot. When \(\eta_{i}(t)=0\), \(i\) goes into a sleep state after processing the tasks in its task queue and does not accept any tasks during a time slot \(t\). In wireless networks, users must connect to a base station (BS) to use communication services. Therefore, when configuring the MEC system, the base stations are partitioned into non-overlapping groups based on their geographical locations, and one MECS is allocated to each group. We denote the group of base stations assigned to a MECS \(i\) by \(B_{i}=\{BS_{i,1},\ldots,BS_{i,n_{i}}\}\). If a BS \(b\) belongs to \(B_{i}\), MECS \(i\) is called the primary MECS of \(b\). If a BS \(b\
otin B_{k}\), MECS \(k\) is said to be the secondary MECS of \(b\). When users send tasks to a BS \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Meaning** \\ \hline \(\mathcal{M}\) & Set of MECSs in the system \\ \hline \(B_{i}\) & Set of base stations (BSs) whose primary MECS is MECS \(i\) \\ \hline \(w_{i,b}(t)\) & Amount of workload newly offloaded through BS \(b\in B_{i}\) during a time slot \(t\) \\ \hline \(\eta(t)\) & MECS status vector for a time slot \(t\) \\ \hline \(Q_{i}(t)\) & Workload of tasks in the task queue of a MECS \(i\) at the start of a time slot \(t\) \\ \hline \(\lambda_{i}(t)\) & Workloads newly assigned to MECS \(i\) during a time slot \(t\) \\ \hline \(\tilde{\lambda}_{i}(t)\) & Workload newly generated in the service area of \(B_{i}\) during a time slot \(t\) \\ \hline \(\tilde{\lambda}_{i}(t)\) & Predicted \(\tilde{\lambda}_{i}(t)\) \\ \hline \(\tilde{\lambda}_{i,j}(t)\) & Workload assigned to active MECS \(i\) due to MECS \(j\) being in sleep mode \\ \hline \(\delta(t)\) & Maximum service delay that MECS \(i\) can provide during a time slot \(t\) \\ \hline \(\zeta(t)\) & Energy consumed by MEC system during a time slot \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations used. Figure 1: System model. \(b\in B_{i}\), \(b\) offloads the task to its primary MECS \(i\) if \(i\) is in the active state.