# Optimal Mobility-Aware Wireless Edge Cloud Support for the Metaverse
## 3 System Model
### Latency, Energy Consumption and Quality of Perception Experience
When finding the target AROs during matching, their pointers, included in foreground interactions, should also be transferred to the metaverse for updating. Finally, the final frame integrating the model and target AROs are transmitted back to the user. Hence, the overall transmission delay for each user after processing using the functions can be written as  
\[ \begin{split}\sum_{s\in\mathbf{S_{r}}}\sum_{j\in\mathbf{M},j \
eq j,}(C_{jA(f(r))}+C_{jA(k)})p_{sj}+\\ (C_{A(f(r))f(r)}+\sum_{k\in\mathbf{K}}C_{A(k)k}u_{f(r)k})\end{split}  \tag{17}\]  
Note that the product of decision variables, \(p_{sj}h_{rl}^{s}\), \(p_{sj}y_{rj}\) and \(p_{sj}h_{rl}^{s}y_{rj}\), creates a non-linearity. Observe that \(p_{sj}h_{rl}^{s}\) and \(p_{sj}y_{rj}\) appear directly while the product \(p_{sj}h_{rl}^{s}y_{rj}\) appears in \(W_{rj}y_{rj}\), which represents the execution of the matching function at the location \(j\) (\(j\in\mathbb{M},j\
eq j_{r}\)). To express the optimization problem in a nominal linear programming setting, we linearize the above expressions via new auxiliary decision variables. To this end, a decision variable \(\alpha_{rsj}\) is introduced as \(\alpha_{rsj}=p_{sj}y_{rj}\) and the constraints should be added, as follows  
\[ \begin{split}\alpha_{rsj}&\leqslant p_{sj},\\ \alpha_{rsj}&\leqslant y_{rj},\\ \alpha_{rsj}&\geqslant p_{sj}+y_{rj}-1\end{split}  \tag{18}\]  
Similarly, a new decision variable \(\beta_{rslj}\) is introduced as \(\beta_{rslj}=p_{sj}h_{rl}^{s}\) and the constraints should be added as follows  
\[ \begin{split}\beta_{rslj}&\leqslant p_{sj},\\ \beta_{rslj}&\leqslant h_{rl}^{s},\\ \beta_{rslj}&\geqslant p_{sj}+h_{rl}^{s}-1\end{split}  \tag{19}\]  
The constraints in (6) can be rewritten as follows,  
\[h_{rl}^{s}\leqslant\sum_{j\in\mathbf{M}}\beta_{rslj},\ \forall r\in\mathbf{R},\ \forall s\in\mathbf{S_{r}},\ \forall l\in\mathbf{L_{rs}} \tag{20}\]In addition, it is worth pointing out that for the decision variable \(p_{sj}\), the following holds: \(p_{sj}=p_{sj}^{2}\). Therefore, we have \(p_{sj}h_{ri}^{s}y_{rij}=\alpha_{rsj}\beta_{rslj}\). Hence, a decision variable \(\lambda_{rslj}\) is defined as \(\lambda_{rslj}=\alpha_{rsj}\beta_{rslj}\) and the following set of constraints are added  
\[\lambda_{rslj} \leqslant\alpha_{rsj}, \tag{21}\] \[\lambda_{rslj} \leqslant\beta_{rslj},\] \[\lambda_{rslj} \geqslant\alpha_{rsj}+\beta_{rslj}-1\]  
Hence, the product \(W_{rj}y_{rj}\) can be rewritten as follows,  
\[\frac{\omega_{\varrho}(F_{er}y_{rj}+\sum_{l\in\mathbf{L_{ns}}}\sum_{\mathbf{ \mathbf{  
The energy efficiency of the system during each service time slot is measured by the production of its total power and running time. The server total power consists of the transmission power and CPU processing power at target ECs. Denote the required CPU processing power of the user \(r\) at the node \(j\) as \(P_{rj}^{cpu}\) and the CPU-chip architecture coefficient as \(k_{0}\) (e.g., \(10^{-18}\)) [5]. Then, the power at the EC can be achieved through \(k_{0}(f_{V}^{j})^{2}\) (J/s), based on measurements in [27; 28]. Noticing that for both conditions, the background contents are processed at servers, the consumed processing time of the server is  
\[T_{cpu}=\sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M},j\
eq m_{r}}V_{rj}x_{rj}+ \sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M}}W_{rj}y_{rj} \tag{27}\]  
The wireless transmission happens regardless of whether foreground interactions are being executed at terminals. Since the selected data rate is \(ge_{rg}\), the wireless transmission time is  
\[T_{trans}=\sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M},j\
eq j,}\frac{F_{r\eta r} ^{fore}}{ge_{rg}}+\sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M},j=j,}\frac{F_{ \sigma r}}{ge_{rg}} \tag{28}\]  
Finally, the total consumed energy at the server side can be written as follows,  
\[