# Optimal Mobility-Aware Wireless Edge Cloud Support for the Metaverse
## 3 System Model  
### Multirendering in Metaverse AR  
For a given wireless network topology with the set \(\mathbb{M}=\{1,2,\ldots,M\}\), we denote the available locations, including available edge clouds and terminals. Assuming that each user makes a single request, the corresponding MAR service requests defined by \(r\in\mathbb{R}\) in the metaverse region generated by mobile users could be equipped with MAR devices. Request \(r\) emerges from network location \(f(r)\). It represents the initial access router, to which this user is firstly connected. For each request, the user terminal could also be viewed as a valid location to cache or to process information locally. Thus, we define with \(j_{r}\in\mathbb{M}\) the terminal sending the request \(r\). The terminals are brought into consideration in the following formulation. Defining the location with constraint \(j\in\mathbb{M},j\
eq j_{r}\) could only enable the ECs. Clearly, for the OptimNT scheme, this works for all locations. However, in the following formulation for the OptimT scheme, we force the storage-intensive functions to be executed only on the ECs, while the computing-intensive ones could also be hosted at the end terminals. During the mobility event, a user could move to different potential destinations \(k\in\mathbb{K}\) (i.e., changing of the anchoring point). Hereafter, and without loss of generality, we only accept adjacent access routers as available destinations in the mobility event. A series of metaverse regions are set on ECs to interact with users. The corresponding metaverse region serving the user can be found through functions \(A(f(r)),A(k)\). As explained earlier, each metaverse region is pre-deployed on a server close to the mobile network and its distance to an EC is also predefined. In this paper, as already suggested, a set of AROs is assumed to be embedded across the different background metaverse region models and is defined as \(\mathbb{N}=\{1,2,\ldots,N\}\). A set \(\mathbb{S}_{r}=\{1,2,\ldots,S\}\) is defined for the multiple rendering of the available metaverse region model to each user. Thus, we denote the decision variable \(p_{sj}\) for pre-caching a metaverse region model \(s\in\mathbb{S}_{r}\) at the EC \(j\left(j\in\mathbb{M},j\
eq j_{r}\right)\). The subset \(\mathbb{L}_{rs}\) represents the target AROs required by the user \(r\) in the related model \(s\in\mathbb{S}_{r}\) and the size of each target ARO \(I\in\mathbb{L}_{rs}\) is denoted as \(O_{I}\). Lastly, the decision variable \(h^{s}_{rl}\) is brought in for proactively caching an ARO required by a request, \(r\). Based on the above, the decision variables \(p_{sj}\) and \(h^{s}_{rl}\) can be defined as follows  
\[p^{j}= \begin{cases}1,\,\text{if rendering the related model $s$ at node $j$,}\\ 0,\,\text{otherwise.}\end{cases}  \tag{1}\]  
\[h^{s}_{rl}= \begin{cases}1,\,\text{if ARO $l$ required by request $r$ embedded}\\ \,\,\text{in the model $s$ is cached,}\\ 0,\,\text{otherwise.}\end{cases}  \tag{2}\]  
Furthermore, the additional set of constraints needs to be satisfied,  
\[\sum_{r\in\mathbf{R}}h^{s}_{rl}\leqslant 1,\,\,\forall j\in\mathbf{M},j\
eq j _{r},\,\,\forall s\in\mathbf{S_{r}},\,\,\forall l\in\mathbf{L_{rs}} \tag{3}\]  
\[\sum_{s\in\mathbf{S_{r}}}\sum_{l\in\mathbf{L_{rs}}}h^{s}_{rl}\geqslant 1,\,\, \forall r\in\mathbf{R} \tag{4}\]  
\[\sum_{j\in\mathbf{M},j\
eq j_{r}}p_{sj}\geqslant h^{s}_{rl},\,\,\forall r\in \mathbf{R},\,\,\forall s\in\mathbf{S_{r}},\,\,\forall l\in\mathbf{L_{rs}} \tag{5}\]  
\[h^{s}_{rl}\leqslant l^{s}_{rl}\sum_{j\in\mathbf{M},j\
eq j_{r}}p_{sj},\,\, \forall r\in\mathbf{R},\,\,\forall s\in\mathbf{S_{r}},\,\,\forall l\in\mathbf{ L_{rs}} \tag{6}\]  
Constraints in (3) force each ARO to be pre-cached at most once in a related model. Constraints (4) ensure that a valid request consists of at least one model and an embedded ARO.