# MEC Server Sleep Strategy for Energy Efficient Operation of an MEC System
## 3 System Model  
In this paper, we consider an MEC system in a wireless network composed of a set of base stations (BSs), a set of MECSs, and a remote controller. We assume that each BS is located with an MECS. Henceforth, we will use MECS and BS interchangeably unless stated otherwise. We divide the system time into a series of time slots with equal length \(\Delta_{t}\) and consider a discrete time controller. Compared with a cloud server, an MECS is resource constrained in terms of the computing power and the memory size. Therefore, an MECS cannot host all the services that a cloud server has at the same time. Since the issue of service caching in an MEC system is dealt in [2; 3; 4], in this paper, we focus on MECS sleep control by assuming that an MECS contains the necessary services for processing tasks offloaded to it. In other words, tasks offloaded to an MEC system is served not by a cloud server but by an MECS. We denote the set of MECSs in the system as \(\mathcal{M}\) and the task processing capacity of an MECS \(i\) as \(F_{i}\). We also denote the service region of an MECS \(i\) as \(S_{i}\). If there is any overlap between \(S_{i}\) and \(S_{j}\), MECS \(i\) and MECS \(j\) are neighboring MECSs. We denote the set of neighboring MECSs of an MECS \(i\) as \(\mathcal{M}_{i}\). A user \(u\) offloads its task to the nearest MECS, which is called the serving MECS of \(u\). We denote the set of users in \(S_{i}\) during a time slot \(t\) as \(U_{i}(t)\). Each MECS has a task queue to accommodate the offloaded tasks. We denote the task queue length of an MECS \(i\) at the beginning of a time slot \(t\) as \(Q_{i}(t)\). We introduce a control variable \(a_{i}(t)\) that indicates the operation mode of an MECS \(i\) during a time slot \(t\). Specifically, when \(a_{i}(t)=0\), an MECS \(i\) is in a sleep mode during a time slot \(t\). On the contrary, \(a_{i}(t)=1\) represents the case where an MECS \(i\) is in an active mode during a time slot \(t\). We denote the number of tasks generated from the users in \(S_{i}\) during a time slot \(t\) as \(\hat{\lambda}_{i}(t)\). If \(a_{i}(t)=1\), \(\hat{\lambda}_{i}(t)\)s are offloaded to their active serving MECS \(i\). However, if \(a_{i}(t)=0\), they are offloaded to active neighbors of the MECS \(i\). Therefore, if we denote the number of tasks newly offloaded to an MECS \(i\) during a time slot \(t\) as \(\lambda_{i}(t)\), it is zero if \(a_{i}(t)=0\). In contrast, when \(a_{i}(t)=1\), \(\lambda_{i}(t)\) becomes the sum of \(\hat{\lambda}_{i}(t)\) and the tasks offloaded to \(i\) from the sleeping neighbors of the MECS \(i\). In other words, if we denote the amount of tasks offloaded to an active MECS \(i\) from its sleeping neighbor \(j\) during a time slot \(t\) as \(\hat{\lambda}_{j\to i}(t)\), \(\lambda_{i}(t)=\hat{\lambda}_{i}(t)+\sum_{j\in\mathcal{M}_{i}}(1-a_{j}(t)) \hat{\lambda}_{j\to i}(t)\). Then, the dynamics of the task queue in an MECS \(i\) is described as  
\[Q_{i}(t+1)=\max\{0,Q_{i}(t)+\lambda_{i}(t)-F_{i}\}. \tag{1}\]  
Bounding the completion time for the tasks offloaded to an MEC system is crucial for delivering a satisfactory service to users. Since we assume that each task is served by an MECS once it is offloaded to an MEC system, the time spent to complete a task in an MEC system is determined by the task queuing delay and the task service delay in a serving MECS, which depends on the service scheduling policy, task size, and CPU capacity. We assume that the task service delay is constant because modern servers use dynamic frequency scaling. If we assume that each MECS processes tasks in a first-in, first-out (FIFO) manner, the queuing delay is proportional to the queue length. As shown in Equation (1), the change in the task queue length in an MECS \(i\) during a time slot \(t\) is determined by the queue length at the beginning of the time slot \(t\), the amount of new tasks entering during the time slot \(t\), and the service capacity of an MECS \(i\). Among these factors, \(\lambda_{i}(t)\) is unknown at the start of the time slot.