## 3 System Model and Problem Formulation
### Joint Problem Formulations  
In accordance with the node service model, let us define the computational resource allocation variable \(y_{mn}\in\Re^{+}\), which signifies the computational resources assigned by service node \(n\) to user \(m\). This paper presumes a computational model that entails simultaneous data processing and transmission; the service node transmits pre-processed data while continuing computations. Therefore, in this paper, \(y_{mn}\) denotes the speed at which transcoding services operate, i.e., the rate of output of computational results per second. If \(y_{mn}<r_{mn}\), part of the available data link capacity remains unused; conversely, if \(y_{mn}>r_{mn}\), user \(m\) may not receive data from service node \(n\) promptly, degrading the user's quality of experience. To enhance the efficiency of logical link utilization and improve the user's vMOS, the following constraint should be adhered to:  
\[\sum_{p_{mn}^{k}\in\mathcal{P}_{mn}}r_{mn}^{k}=h_{mn}y_{mn},\quad\forall m\in \mathcal{M},n\in\mathcal{N} \tag{5}\]  
To fulfill the computational demands for varying video resolution quality levels required by distinct users, it is essential for the network to allocate the correct amount of computational resources. Moreover, to optimize the utilization of these resources at the service nodes, the computational resources across all service nodes must align with the transcoding needs specific to each user. The computational resources necessary for transcoding are directly linked to the data rate for the chosen resolution level by the user. To ensure this alignment, the following condition must be satisfied:  
\[\sum_{g=1}^{G}z_{gm}c_{g}=\sum_{n\in\mathcal{N}}y_{mn},\quad\forall m\in \mathcal{M},n\in\mathcal{N} \tag{6}\]  
Besides the limits on link power capacity discussed earlier, network design must also take into account the following physical resource constraints: restrictions on wired connections, wireless connections, and the computational capacity at service nodes. These factors have a significant impact on both network performance and reliability. Specifically, the following conditions must be met:  
\[\sum_{p_{mn}^{k}\in\mathcal{P}_{l}}a_{l}r_{mn}^{k}\leq B_{l},\quad\forall l\in \mathcal{L}_{wl} \tag{7}\]  
\[\sum_{p_{mn}^{k}\in\mathcal{P}_{l}}a_{l}r_{mn}^{k}\leq\hat{R}_{l}^{\max},\quad \forall l\in\mathcal{L}_{wl} \tag{8}\]  
\[\sum_{m\in\mathcal{M}}y_{mn}\leq C_{n},\quad\forall n\in\mathcal{N} \tag{9}\]  
The goal of this study is to enhance the network's overall efficiency, with the average vMOS for users chosen as the utility function. Taking into account the previously outlined constraints, the problem, denoted as **P0**, is formulated as follows:  
\[\textbf{P0}:\quad\min_{\textbf{Z,X,Y,R,P}} -\frac{1}{|\mathcal{M}|}\sum_{m\in\mathcal{M}}\sum_{g=1}^{G}s_{g} z_{gm}\] (10) s. t. \[C1:\quad\sum_{g=1}^{G}z_{gm}=1,\quad\forall m\in\mathcal{M} \tag{11}\] \[C2:\quad\sum_{j\in\mathcal{J}}x_{mj}=1,\quad\forall m\in \mathcal{M}\] (12) \[C3:\quad\sum_{p_{mn}^{k}\in\mathcal{P}_{mn}}r_{mn}^{k}=h_{nm}y_{ mn},\quad\forall m\in\mathcal{M},n\in\mathcal{N}\] (13) \[C4:\quad\sum_{g=1}^{G}z_{gm}c_{g}=\sum_{n\in\mathcal{N}}y_{mn}, \quad\forall m\in\mathcal{M}\] (14) \[C5:\quad\sum_{p_{mn}^{k}\in\mathcal{P}_{m}}a_{mj}r_{mn}^{k}=x_{mj }\sum_{g=1}^{G}z_{gm}v_{g},\quad\forall m\in\mathcal{M},j\in\mathcal{J}\] (15) \[C6:\quad EPFD_{j}\leq EPFD_{ht},\quad\forall j\in\mathcal{J}_{geo}\] (16) \[C7:\quad\sum_{m\in\mathcal{M}}y_{mn}\leq C_{n},\quad\forall n\in \mathcal{N}\] (17) \[C8:\quad\sum_{l\in L_{2}}2^{\hat{R}}\leq P_{n},\quad\forall n\in \mathcal{N}\] (18) \[C9:\quad\sum_{p_{mn}^{k}\in\mathcal{P}_{l}}a_{l}r_{mn}^{k}\leq B_ {l},\quad\forall l\in\mathcal{L}_{ind}\] (19) \[C10:\quad\sum_{p_{mn}^{k}\in\mathcal{P}_{l}}a_{l}r_{mn}^{k}\leq \hat{R}_{l}^{\max},\quad\forall l\in\mathcal{L}_{wl}\] (20) \[C11:\quad z_{gm},x_{mj}\in\{0,1\},r_{mn}^{k},p_{l},y_{mn}\in \Re^{+} \tag{21}\]