## 2 System Model and Problem Formulation  
We consider a heterogeneous edge computing network as shown in Figure 1, where each edge network includes \(P\) edge computing nodes, an edge server node and a remote data center. The edge computing nodes have heterogeneous computing processing and communication capabilities. Edge server as the main node, mainly responsible for computing task scheduling, with sufficient power. We consider that tasks in the heterogeneous edge computing network have different data type computing requirements and are independent and random. It needs to be noted that different data types correspond to different processing capabilities for our computing nodes. For example, devices equipped with graphics processing GPUs usually have higher processing power for video data. For devices with multi-core CPUs, matrix operations are often more efficient. The computing task is transmitted from the remote data center to the edge server. The edge server is assigned to the edge computing node to complete the computing task and finally returns to the data center. According to the energy consumption model of the edge computing network, when the edge computing node \(p\) performs the computing task \(m\), the required computing energy consumption can be expressed as  
\[E_{p,m}^{comp}=e_{p,m}^{comp}u_{p,m}, \tag{1}\]  
where \(e_{p,m}^{comp}\) represents the unit computational energy consumption when the edge computing node \(p\) performs task \(m\), and \(u_{p,m}\) represents the data size requirement for task \(m\). Assuming that after edge computing \(p\) completes task \(m\), the size of the collected packet is \(L_{p,m}\), considering the transmission distance as \(d\), we can give the communication energy consumption required by edge computing \(p\) to perform task \(m\) as  
\[E_{p,m}^{comm}=\left\{ \begin{array}{ll}L_{p,m}E_{elec}+L_{p,m} \varepsilon_{sf}d^{2},&d\leq d_{0}\\ L_{p,m}E_{elec}+L_{p,m}\varepsilon_{mp}d^{2},&d\geq d_{0}\end{array} \right., \tag{2}\]  
where \(E_{elec}\) represents the energy consumption required to send or receive unit data, \(\varepsilon_{sf}\) and \(\varepsilon_{mp}\) represents the free space model magnification and the multipath transmission model magnification, respectively. The transmission distance is related to the wireless channel. It can be known that within a certain distance, inter-device communication usually has a good wireless channel. As the distance gradually increases, the impact of multipath effects on the wireless channel begins to become severe. At this time, the parameters of multipathtransmission should be used to describe the wireless channel. For the transmission distance threshold \(d_{0}\), we have  
\[d_{0}=\sqrt{\frac{\varepsilon_{f_{s}}}{\varepsilon_{mp}}}. \tag{3}\]  
In order to tackle the resource scheduling problems caused by the heterogeneous characteristics of edge computing network, we introduce software-defined network to realize the separation of control layer and data layer. The idea of separating data layer and control layer is embodied in software defined network. For the network we consider, the corresponding control layer of the resource scheduling strategy, the processing of computing tasks and the corresponding data layer of transmission. The control layer is independent of the data layer, allowing the network to have better compatibility and scalability. At the same time, the architecture can also realize the dynamic adjustment of the network, improve the overall network efficiency and make the network more intelligent. Such a software-defined heterogeneous edge computing network architecture effectively increases network flexibility and enables more efficient resource management. Specifically, the software-defined heterogeneous edge computing network architecture is shown in Figure 2. The edge server with enough power acts as a control layer node. The edge server node in control layer can realize the centralized deployment, and also can realize the separation of control layer and data layer.