# Joint Content Recommendation and Delivery in Mobile Wireless Networks with Outage Management
## 2 System Model
### Problem Formulation  
We consider the decision problem of joint content recommendation and delivery at the content server. Apart from tracking the users' interests on content files, the content server also periodically monitors the user and channel dynamics reported from the BS. Based on such information, a decision should be made to recommend to each user a list of \(N\) contents. To manage the user QoE, the recommendation algorithm should guarantee that in the worst case (when all users each request a piece of content from their lists at the same time), the system capacity is able to satisfy the user requests within a time constraint \(T_{s}\). At the system level, the service QoE is evaluated by two matrices. One is the sum user interests over recommended contents and the other is the outage probability in the worst case scenario. There is a tradeoff between these two matrices. As a result, the basic idea underpinning our problem formation is to maximize the sum user interests, under the condition that the outage probability is constrained by a predefined parameter. However, in practice, the exact outage probability is difficult to calculate. Therefore, as an indirect approach to outage management, we propose two alternative methods to constrain the sizes of recommended files according to the available channel capacity. The first approach constrains the maximum file size to give:  
\[\max_{f=1\dots F}\{x_{uf}/f_{f}\}\leq\delta*B*\sum_{k=1}^{K}c_{uk}*\rho_{uk}*T_ {s}\qquad\forall u \tag{9}\]  
where \(x_{uf}\in\{0,1\}\) is the decision variable that takes binary values. When \(x_{uf}\) is one, it means that content \(f\) is recommended to user \(u\), otherwise \(x_{uf}\) is zero. Here, parameter \(\delta(1\leq\delta)\) is a parameter used to control the outage probability. When \(\delta=1\), this means that the recommendation is conservative in the sense that the maximum size of all \(N\) recommended files will not be greater than the estimated channel capacity allocated to the user. When \(\delta\) is larger than one, this means that a certain outage is allowed in the system. The maximum file size constraint in (9) is intuitive, but nonlinear. To simplify the problem, it is desirable to have a linear constraint. To this end, we propose another constraint as follows:  
\[\frac{1}{N}\sum_{f=1}^{F}x_{uf}/f_{f}\leq\delta*B*\sum_{k=1}^{K}c_{uk}*\rho_{uk }*T_{s}\qquad\forall u. \tag{10}\]  
This constraint limits the average file size in the recommendation list instead of the maximum file size. Applying the above constraints, the problem of joint content recommendation and delivery is formulated as:  
maximize \[\sum_{u=1}^{U}\sum_{f=1}^{F}r_{uf}x_{uf}\] (11) subject to Equation (9) or (10) \[B*\sum_{k=1}^{U}\sum_{k=1}^{K}\frac{f(c_{uk})}{\alpha_{uk}^{2}}* \rho_{uk}\leq P_{T}\] \[\sum_{f=1}^{F}x_{uf}=N\] \[\sum_{u=1}^{U}\rho_{uk}\leq 1\] \[x_{uf}\in\{0,1\},\rho_{uk}\in\{0,1\}.\]  
In this problem formulation, the objective is to maximize the total user interest. The first constraint corresponds to the capacity outage control; the second constraint reflects the total BS transmit power limit \(P_{T}\); the third constraint limits the number of recommended files to be \(N\); the fourth constraint implies orthogonal subcarrier allocation. The decision variables in our problem are \(x_{uf}\), \(\rho_{uk}\) and \(c_{uk}\). This means that the optimization is jointly performed over content recommendation and content delivery (i.e., channel/power/bit allocations in OFDM-based wireless communications systems). If we do not consider the aspects of content delivery, the above problem reduces to the traditional recommendation problem given by:  
\[ \begin{array}{ll}\text{maximize}&\sum_{u=1}^{U}\sum_{f=1}^{F}r_{uf}x_{uf}\\ \text{subject to}&\sum_{f=1}^{F}x_{uf}=N\\ &x_{uf}\in\{0,1\}.\end{array}  \tag{12}\]  
The traditional recommendation is able to recommend the most interested contents to users, but often at the cost of high capacity outage probability and hence degraded user experience.