# Optimal Proactive Caching for Multi-View Streaming Mobile Augmented Reality
## 3 System Model
### Mathematical Programming Formulation
#### 3.2.2. Latency and Preference  
Based on the above derivations and in line with [8], the service latency can be written as follows,\[L= (21)+\sum_{r\in\mathbf{R}}\sum_{i\in\mathbf{M}}(C_{f(r)i}+V_{ri})x_{ri}+ \tag{22}\] \[\sum_{r\in\mathbf{R}}\sum_{i\in\mathbf{M}}\sum_{j\in\mathbf{M}}( \frac{\omega(F_{\text{er}}y_{rj}+\sum_{l\in\mathbf{L_{ns}}}\sum_{s\in\mathbf{S_{ r}}}\lambda_{rsl}Q_{l})}{f_{V}^{l}}+\] \[C_{ij}\xi_{rij}+C_{jf(r)}y_{rj}+\psi_{rj}D)+\]  
\[\sum_{s\in\mathbf{S_{r}}}\sum_{i\in\mathbf{M}}\sum_{j\in\mathbf{M}}C_{ij}\alpha_ {rsj}+\sum_{s\in\mathbf{S_{r}}}\sum_{r\in\mathbf{R}}\sum_{j\in\mathbf{M}}\sum_ {k\in\mathbf{K}}C_{jk}u_{f(r)k}p_{sj}\]  
\(L_{max}\) here denotes the maximum allowed service latency. Thus, we have,  
\[\frac{L}{L_{max}}\in[0,1] \tag{23}\]  
The overall preference of view streams and embedded AROs is measured by the summation of probabilities of the chosen view streams and AROs. It is denoted as \(Q\) and can be written as follows,  
\[Q =\sum_{s\in\mathbf{S_{r}}}\sum_{j\in\mathbf{M}}p_{sj}t_{s}+\sum_{r \in\mathbf{R}}\sum_{s\in\mathbf{S_{r}}}\sum_{j\in\mathbf{M}}\sum_{l\in \mathbf{L_{ns}}}p_{sj}h_{ri}^{s}t_{l} \tag{24}\] \[=\sum_{s\in\mathbf{S_{r}}}\sum_{j\in\mathbf{M}}p_{sj}t_{s}+\sum_{ r\in\mathbf{R}}\sum_{s\in\mathbf{S_{r}}}\sum_{j\in\mathbf{M}}\sum_{l\in \mathbf{L_{ns}}}\beta_{rslj}t_{l}\]  
similarly, \(Q_{max}\) here denotes the maximum allowable preference of view streams and embedded AROs. Hence, for the preference of view streams and embedded AROs, we also have,  
\[\frac{Q}{Q_{max}}\in[0,1] \tag{25}\]  
The weight parameter is denoted as \(\mu\in[0,1]\), and the joint optimization problem can eventually be written as follows,  
\[min\ \mu\frac{L}{L_{max}}-(1-\mu)\frac{Q}{Q_{max}}\] (26a) s.t. \[z_{rj}=1-q_{rj},\ \forall j\in\mathbf{M},r\in\mathbf{R} \tag{26b}\] \[\sum_{r\in\mathbf{R}}(x_{rj}+y_{rj})\leq\Delta_{j},\forall j\in \mathbf{M}\] (26c) \[\sum_{j\in\mathbf{M}}x_{rj}=1,\forall r\in\mathbf{R}\] (26d) \[\sum_{j\in\mathbf{M}}y_{rj}=1,\forall r\in\mathbf{R}\] (26e) \[\xi_{rij}\leq x_{ri},\ \forall r\in\mathbf{R},i,j\in\mathbf{M}\] (26f) \[\xi_{rij}\leq y_{rj},\ \forall r\in\mathbf{R},i,j\in\mathbf{M}\] (26g) \[\xi_{rij}\geq x_{ri}+y_{rj}-1,\ \forall r\in\mathbf{R},i,j\in \mathbf{M}\] (26h) \[\psi_{rj}\leq z_{rj},\ \forall r\in\mathbf{R},j\in\mathbf{M}\] (26i) \[\psi_{rj}\leq y_{rj},\ \forall r\in\mathbf{R},j\in\mathbf{M}\] (26j) \[\psi_{rj}\geq z_{rj}+y_{rj}-1,\ \forall r\in\mathbf{R},j\in \mathbf{M}\] (26k) \[x_{rj},y_{rj},p_{sj},h_{ri}^{s},z_{rj},q_{j}\in\{0,1\},\] \[\alpha_{rsj},\beta_{rslj},\lambda_{rslj},\psi_{rj},\xi_{rij}\in\{0,1\},\] \[\forall r\in\mathbf{R},j\in\mathbf{M},l\in\mathbf{L_{rs}},s\in \mathbf{S_{r}}\] (26l) \[(3),\ (4),\ (5),\ (11),\ (12),\ (15),\ (16),\ (17),\ (18)\]As mentioned earlier, the constraint (26b) together with constraints (3) to (5), (11), (12) and (17) reveal the relation between pre-caching decisions and the cache miss/hit for each request [8]. The constraint (26c) is the virtual machine limitation while (26d) and(26e) guarantee the once execution of each function of a request at a single server as explained in [8]. The constraints (15), (16), (18) and (26f) to (26k) are auxiliary and required to solve the product of decision variables for linearization. Key variables defined in the formulation are shown with their description in the following Table 1.