# Optimal Mobility-Aware Wireless Edge Cloud Support for the Metaverse
## 3 System Model
### Latency, Energy Consumption and Quality of Perception Experience
\begin{split} E_{server}&=\sum_{r\in\mathbf{R}}\sum _{j\in\mathbf{M}}(P_{rj}^{trans}T_{tran}+P_{rj}^{cpu}T_{cpu})\\ &=\sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M}}(\frac{N_{j}+\sum_{i \in\mathbf{M},i\
eq j}P_{i}H_{ri}^{2}d_{ri}^{-a}}{H_{rj}^{2}d_{ri}^{-a}}(2^{ \frac{ge_{rg}}{k_{j}}}-1))\\ &(\sum_{j\
eq j,}\frac{F_{\eta r}^{fore}}{ge_{rg}}+\sum_{j=m_{r}} \frac{F_{\sigma r}}{ge_{rg}})\\ &\quad+\sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M}}k_{0}(f_{V}^{ j})^{2}(W_{rj}y_{rj}+\sum_{j\
eq j_{r}}V_{rj}x_{rj})\end{split}  \tag{29}\]  
Regarding the terminal side, in this paper, we follow the Amdahl's law to model the energy consumption, in which the potential speedup of potential parallel computations is considered in the function [29]. In this paper, the metaverse AR functions work by serial and, for simplicity, the parallel portion is assumed to be zero like in [17]. As mentioned earlier, the dynamic foreground interactions, including some highly used AROs, could be proactively cached at terminals with the matching function. The 3D background model, on the other hand, is much larger and might serve multiple users in a region. Hence, it is not recommended to store or process this at the terminal. In addition, the metaverse application should not exceed a certain portion of the whole terminal CPU resources, so that other functionalities can work properly [17]. Denoting the consumed portion as \(\Gamma_{r}\in[30\%,50\%]\)[17], then, \(\frac{V_{rj}x_{rj}}{\Gamma_{r}}\) means processing metaverse AR functions at the terminal requires a longer time. The energy consumption of terminals can be written as follows,  
\[ \begin{split} E_{terminal}=& P_{terminal}T_{terminal}\\ =&\sum_{r\in\mathbf{R}}\sum_{j=j,}k_{0}(f_{V}^{j})^{ 2}\frac{V_{rj}x_{rj}}{\Gamma_{r}}\end{split}  \tag{30}\]  
Finally, the overall system energy consumption is \(E=E_{server}+E_{terminal}\). \(E_{max}\) represents the maximum possible energy consumption of the system. It also has \(\frac{E}{E_{max}}\in[0,1]\). SSIM is applied to reveal the quality of perception experience. In this paper, the video coding scheme (e.g., H.264) and frame resolution (e.g., 1280 \(\times\) 720) are assumed as pre-defined [10]. Then, SSIM is mainly affected by data rate and a concave function can be applied to reveal the relation between them [10]. Hence, the set of SSIM values for each ARO under corresponding data rates can be denoted as \(\mathbb{SSIM}_{l},l\in\mathbb{L}_{r}\). The overall quality of perception experience, \(Q\), can be written as follows,  
\[Q=\sum_{r\in\mathbf{R}}\sum_{l\in\mathbf{L}_{r}}\sum_{g\in\mathbf{G}}\sum_{c\in \mathbb{SSIM}_{l}}e_{rg}c \tag{31}\]  
To maintain the user experience above an acceptable level, the perception-quality constraint can be added, as follows,  
\[\frac{Q}{Q_{max}}\geq Q_{bound} \tag{32}\]  
where \(Q_{max}\) is the maximum quality through selecting max allowable data rate and storing as many AROs as possible. Using a weighting parameter \(\mu\in[0,1]\), the bi-objective optimization problem can be written as follows,  
\[min\ \mu\frac{L}{L_{max}}+(1-\mu)\frac{E}{E_{max}}\] (33a) s.t.