# Optimal Proactive Caching for Multi-View Streaming Mobile Augmented Reality
## 3 System Model
### Mathematical Programming Formulation  
In a similar manner with [8], computational-intensive and storage-intensive MAR functionalities are defined as \(\eta\) and \(\varrho\), respectively. In addition, the corresponding execution location for a functionality is denoted as \(x_{ri}\) and \(y_{ri}\), respectively [8]. The starting location of the request \(r\) is the access router where this user is initially connected to; this initial location is defined as \(f(r)\). A user moves to a destination \(k\in\mathbb{K}\) in the case of a mobility event (i.e., changing the point of attachment). In this paper, and without loss of generality, only adjacent access routers can be regarded as allowable destinations in the mobility event. Hence, \(u_{f(r)k}\in[0,1]\) is defined to represent the probability of a user moving from the initial location to an allowable destination, where adjacent servers \(\{f(r),k\}\subset\mathbb{M}\). Such probabilities can also be learned from historical data, which are readily available from mobile operators. The size of uploaded frames with target AROs used in the Object Detection function is denoted as \(F_{irr}\) and the size of pointers used for identifying target AROs is denoted as \(F_{\varnothing r}\)[8; 10]. During the matching process, the target AROs are possibly not pre-cached in the local cache and such case is known as a \"cache miss\" (otherwise there is a \"cache hit\"). Whenever confronted with a cache miss, the request is redirected to a core cloud deeper in the network and suffers from an extra latency \(D\) as penalty. At first, a joint optimization scheme considering the balance between the service delay and the preference of view streams and embedded AROs is designed to track the MAR requests in the EC-supported network. The cache hit/miss is captured by the decision variable \(z_{rj}\) and can be written as follows,  
\[z_{rj}= \begin{cases}1,\,\text{if}\sum_{l\in\mathbf{L}_{\text{rs}}}\sum_{s\in \mathbf{S}_{\mathbf{r}}}p_{sj}h_{rl}^{s}\geqslant L_{rs},\\ 0,\,\text{otherwise}.\end{cases}  \tag{10}\]  
In [8], constraints (10b) and (10d) reveal the cache limitation and the cache hit/miss relation. In this paper, they should be rewritten as follows,  
\[\sum_{r\in\mathbb{R}}\sum_{l\in\mathbf{L}_{\text{rs}}}\sum_{s\in\mathbf{S}_{ \mathbf{r}}}p_{sj}h_{rl}^{s}O_{l}\leq\Theta_{j},\forall j\in\mathbf{M} \tag{11}\]  
\[\sum_{l\in\mathbf{N}}\sum_{s\in\mathbf{S}_{\mathbf{r}}}h_{rl}^{s}+\epsilon \leq L_{rs}+U(1-q_{rj})\,\forall j\in\mathbf{M},r\in\mathbf{R} \tag{12}\]  
where \(\Theta_{j}\) is the cache capacity at node \(j\). In (12), to rewrite the either-or constraint that \(\sum_{l\in\mathbb{N}}\sum_{s\in\mathbf{S}_{\mathbf{r}}}h_{rl}^{s}<L_{rs}\) or \(z_{rj}=1\), we define \(\epsilon\) as a small tolerance value, \(U\) as a large arbitrary number and \(q_{rj}\) as a new decision variable satisfying \(1-q_{rj}=z_{rj}\)[8]. Requiring more pre-cached view streams and embedded AROs in a request naturally lead to an extra burden for the matching function. More specifically, the processing delay of the matching function can be written as,  
\[W_{rj}=\frac{\omega(F_{\varnothing r}+\sum_{l\in\mathbf{L}_{\text{rs}}}\sum_{s \in\mathbf{S}_{\mathbf{r}}}p_{sj}h_{rl}^{s}O_{l})}{f_{V}^{j}} \tag{13}\]  
where \(\omega\) is the computation load, \(f_{V}^{j}\) is the virtual CPU frequency and \(F_{\varnothing r}\) are the size of the uploaded pointers [8; 10]. When finding the target ARO during matching, other view streams containing it should also be transferred to the user. Hence, the delay of sending results back to the user can be written as follows,  
\[\sum_{s\in\mathbf{S}_{\mathbf{r}}}\sum_{i\in\mathbf{M}}\sum_{j\in\mathbf{M}}C_ {ij}y_{ri}p_{sj}+\sum_{s\in\mathbf{S}_{\mathbf{r}}}\sum_{r\in\mathbf{R}}\sum_{ j\in\mathbf{M}}\sum_{k\in\mathbf{K}}C_{jk}u_{f(r)k}p_{sj} \tag{14}\]  
where \(u_{f(r)k}\) is the moving probability from the initial location \(f(r)\) to a potential destination \(k\).