## 2 Materials and Methods
### Preliminary Knowledge
#### 2.1.2 Introduction of Model Components
Thus, it can be expressed as:  
\[\mathrm{e}_{\varphi=2,m}\left(\overrightarrow{u_{m}},\overrightarrow{v_{m}} \right)=\mathrm{P}_{s}+\mathrm{n}(\mathrm{F}_{m})^{2}\mathrm{B}_{m}+\sum_{i= 1}^{N}\mathrm{v}_{m,i}\times\frac{\mathrm{c}\mathrm{B}_{m}}{\mathrm{U}_{m} \left(\overrightarrow{u_{m}},\overrightarrow{v_{m}}\right)}, \tag{9}\]  
In the equation, \(\mathrm{n}\) is the factor of energy consumption, equal to 3.44 \(\times\) 10\({}^{-23}\). The first term in the equation is the standby consumption of the mobile device. The consumption of local computation is represented by the second term. The last term is the uplink transmission consumption, which is equal to the product of transmission power and time. For interactive tasks, generally, the task is processed by local and edge computing at the same time. Therefore, the energy consumption of a task can be expressed as the sum of device standby consumption, local computing consumption and uplink transmission consumption:  
\[\mathrm{e}_{\varphi=3,m}\left(\overrightarrow{u_{m}},\overrightarrow{v_{m}} \right)=\mathrm{P}_{s}+\mathrm{\mu}_{m}\mathrm{n}(\mathrm{F}_{m})^{2}\mathrm{ B}_{m}+\sum_{i=1}^{N}\mathrm{v}_{m,i}\times\frac{(1-\mathrm{\mu}_{m})\mathrm{c} \mathrm{B}_{m}}{\mathrm{U}_{m}\left(\overrightarrow{u_{m}},\overrightarrow{ v_{m}}\right)}, \tag{10}\]  
The meaning of terms in Equation (10) will not be repeated. * User cooperation model User cooperation is used to promote the optimization of wireless networks. In some scenarios, the network will have to process a large number of requests for one or some specific downlink tasks; for example, video-on-demand services on trains and planes, or when a video is popular in a specific region within a certain period of time. In this case, the server can send the same or similar content to multiple users together by collecting specific information in user requests, which can effectively reduce traffic in the network and promote network optimization. This can not only make the model converge faster but can also effectively save communication resources and improve the performance of the optimized network. To solve this problem, a necessary assumption is that the number of popular videos is small; thus, the above problem can be solved by adding a short identifier to the request sent by users. The paper further postulates that the identifier has K bits in total, where k\({}_{1}\) bits indicate the kind of task (uplink, downlink or interactive task). The rest of the k\({}_{2}\) bits are used to represent the index of the specific task. The size of the content transmitted by the user should be:  
\[\text{B}_{\text{m}}=\text{B}_{\text{info}}+\text{K}, \tag{11}\]  
This model brings two benefits to optimization: It can make the best use of network resources and compress the exploration space of the algorithm. The former can significantly reduce the network transmission delay, while the latter can simplify the optimization problem and improve the final results. This paper assumes a network dominated by downlink tasks, and where the number of different tasks is relatively small. Thus, it is unwise to process the same or similar requests for tasks separately, as this will waste precious bandwidth resources. After considering the user cooperation model, these similar tasks can be processed together to achieve a higher transmission rate with the original bandwidth. Let us take two homogeneous downlink tasks as an example to explain this concept. Suppose that two independent users, m1 and m2, submit requests for similar downlink tasks. Recall in Equation (4), if user cooperation is not considered, the maximum transmission delay of two users is expressed as:  
\[\mathrm{D_{m,j}(d_{m,j},\ w_{m,j})}=d_{m,j}W\log_{2}\left(1+\frac{w_{m,j}h_{m, j}}{\sigma^{2}}\right), \tag{2}\]0  
where B\({}_{\text{content}}\) is the size of the transmitted content and D\({}_{\text{m1}}\) and D\({}_{\text{m2}}\) are the transmission rates of the two users.