## 2 Preliminaries
### System Model  
As shown in Figure 1, the system model of the low-latency MEC network we consider is composed of a set of UEs \(\mathcal{U}=\{1,2,\cdots,K\}\) and one MEC server. We discretize time into frames of consistent length. At the beginning of each frame, the task arrives randomly. A task consists of multiple data packets of different sizes. Each packet is stored in a user's queue and then transmitted to the edge server at a later frame. One frame may transmit one or more packets. Once all data packets of a task have been successfully transmitted, the task can start computation and then be used to decide when the computation is finished. Specifically, we discretize time into a set of time frames \(\mathcal{T}=\{1,2,\cdots,T\}\). Each frame is comprised of \(n\) symbols, which are the smallest units required to represent the transmitted data in our communication system. A symbol is not merely a bit, instead, it encapsulates a discrete value or a set of bits according to the modulation scheme used. The duration of each symbol is \(T_{s}\), which is how long it takes to transmit one symbol. Therefore, the total duration of the frame \(T_{f}\) can be represented as  
\[T_{f}=nT_{s} \tag{1}\]  
Each frame can be split into a communication phase and a computation phase. The frame structure is shown in Figure 2. During the transmission phase, data packets from different UEs are transmitted to the edge server via the wireless channels using time-division multiple access. During the computation phase, we consider joint-packet task scenarios. That is, a task is made up of data packets from multiple UEs. A task's computation and decision-making process can only be initiated when all data packets involved have been transmitted (transmission can be distributed across different frames). Considering that the ability to start decision-making for a task depends mainly on whether the last data packet of the task has been successfully transmitted, we utilize a fixed time length for the computation phase. Tasks in which all packets have been fully transmitted can proceed with calculation and decision-making in the computation phase. Data packets may arrive at the beginning of each frame, that is, before the transmission starts. A data packet with packet size \(m_{k,t}\) (in bits) arrives at the UE \(k\) in frame \(t\) with a probability of \(p\), suggesting that the data packet arrivals conform to an On/Off Markov arrival model:  
\[P= \begin{cases}p&\text{packet arrives}\\ 1-p&\text{packet does not arrive}\end{cases}  \tag{2}\]  
Furthermore, for each UE, a data packet is then cached in an independent First-In-First-Out (FIFO) queue. Notably, we consider joint task decisions that require multiple data packets. This suggests that for a specific task, its data packets always arrive at various UEs at the same time (e.g., in an autonomous driving scenario, we can realize a comprehensive perception of the road environment at a specific moment by aggregating the data captured from different sensors at the exact moment). Figure 1: System model of low-latency MEC network. Figure 2: Frame structure. Additionally, due to potential variations in the types of UEs, the sizes of data packets arriving at each UE may differ. We assume that the packet size \(m_{k,t}\) is uniformly distributed across the range\([a,b]\), ensuring that each packet size within this interval is equally likely to occur. The probability density function of the packet size \(m_{k,t}\) is given by:  
\[f(m_{k,t})= \begin{cases}\frac{1}{b-a}&a\leq m_{k,t}\leq b\\ 0&\text{otherwise}\end{cases}  \tag{3}\]  
Channels are considered to experience quasi-static Rayleigh fading. Therefore, as the duration of a frame \(T_{f}\) is shorter than the channel coherent time, the channel fading remains constant within the frame but may vary between the current frame and the next. Channels between different UE and edge servers are considered to be independent.