## 2 System Model
### Computation Task Model  
At time-slot \(t\), the computation task of the IoT device \(k\) is denoted as \(\beta_{k,t}\), which is defined by a tuple \((l_{k,t},c_{k,t},\tau_{k,t})\), where \(l_{k,t}\) denotes the size (in bits) of the task \(\beta_{k,t}\), \(c_{k,t}\) denotes the number of required CPU cycles for computing 1-bit of the task \(\beta_{k,t}\) (i.e. the computational complexity), and \(\tau_{k,t}\) is the latency requirement of the task \(\beta_{k,t}\). Similar to the assumption in [27], we assume that at the beginning of each time-slot \(t\), each IoT device \(k\) has a new task arrival, \(l_{k,t}\) is randomly generated from the set \(\mathcal{L}_{k}=\left\{L_{1}^{(k)},L_{2}^{(k)},\ldots,L_{N_{k}}^{(k)}\right\}\), and the corresponding computational complexity \(c_{k,t}\) belongs the set of complexity \(\mathcal{C}_{k}=\left\{C_{1}^{(k)},C_{2}^{(k)},\ldots,C_{N_{k}}^{(k)}\right\}\). For simplicity, the latency requirement for each task is set to \(\tau_{k,t}=\Delta,\forall k\in\mathcal{K},t\in\mathcal{T}\).  
To facilitate the resource management, a virtual system operator (VSO) is deployed at the BS, which is responsible for collecting the network information (e.g., the channel state information, size of the each IoT device's task, each IoT device's task computational complexity, etc.) and allocating computation resources for the IoT devices. As the computing  
Figure 1: The illustration of the multi-MEC server cellular network.  
powers of the IoT devices are always weak, we assume that the tasks of each IoT device must be entirely offloaded to a certain MEC server for computation through wireless link. The task offloading decision variable of the IoT device \(k\) is denoted as \(\mu_{k,m,t}\in\{0,1\}\), where \(\mu_{k,m,t}=1\) denotes the IoT device \(k\)'s computation task is offloaded to the MEC server \(m\) for execution at time-slot \(t\).  
Let \(\tau_{k,m,t}\) denote the time duration of task offloading from the IoT device \(k\) to the MEC server \(m\) at time-slot \(t\). The offloaded task is processed by the MEC server \(m\) in remaining time duration \(\Delta-\tau_{k,m,t}\). The rate of task data offloaded by the IoT device \(k\) to the MEC server \(m\) at time-slot \(t\) can be expressed as  
\[R_{k,m}[t]=B_{k,m}\text{log}_{2}\bigg{(}1+\frac{h_{k,m,t}p_{k,m,t}}{\sigma_{m} ^{2}}\bigg{)},k\in\mathcal{K},m\in\mathcal{M},t\in\mathcal{T} \tag{1}\]  
where \(B_{k,m}\) denotes the available channel bandwidth between the IoT device \(k\) and the MEC server \(m\); \(p_{k,m,t}\) denotes the transmit power of the IoT device \(k\) at the time-slot \(t\), \(\sigma_{m}^{2}\) is the noise power at the MEC server \(m\). The energy consumption of the IoT device \(k\) at time-slot \(t\) for task offloading is expressed as  
\[E_{m,k}^{o}[t]=p_{k,m,t}\tau_{k,m,t},k\in\mathcal{K},m\in\mathcal{M},t\in \mathcal{T} \tag{2}\]  
The corresponding computation energy consumption of the MEC server \(m\) can be expressed as  
\[E_{m,k}^{c}[t]=\rho_{m}f_{m,k,t}^{3}(\Delta-\tau_{k,m,t}),k\in\mathcal{K},m\in \mathcal{M},t\in\mathcal{T} \tag{3}\]  
where \(f_{m,k,t}\) denotes the CPU frequency of the MEC server \(m\) allocated to the IoT device \(k\)'s task at time-slot \(t\); \(\rho_{m}\) is the effective capacitance coefficient of the MEC server \(m\) at time-slot \(t\), which is determined by the chip architecture [6; 7].