# Joint Content Recommendation and Delivery in Mobile Wireless Networks with Outage Management
## 5 Performance Evaluation Framework
### Approximation of Outage Probability  
Unfortunately, evaluating the system outage according to its real definition in Equation (29) is not only mathematically intractable, but also computationally challenging. To facilitate our performance evaluation, we propose a method that can approximately estimate the transmission outage according to the following equation:  
\[\eta=P\left(\sum_{u=1}^{U}l_{i}^{u}>C_{sys}*T_{s}\approx\sum_{u=1}^{U}R_{u}*T_{s }\right). \tag{30}\]  
Here, the outage probability \(\eta\) is defined as the probability that the overall requested file size is greater than the instantaneous system capacity \(C_{sys}\). This approximation essentially neglects the channel diversity of multiple users and uses a lump-sum capacity \(C_{sys}\) to capture the resource limitation of the system. There exist many methods to calculate the lump-sum capacity \(C_{sys}\). Without loss of generality, we propose to calculate \(C_{sys}\) as the sum capacity of a system that adopts the same radio resource allocation strategy introduced in Section 4.1. In other words, the sum capacity obtained in the capacity estimation phase will be used as the lump-sum system capacity for outage estimation.  
Now, we proceed to investigate fast algorithms to evaluate the system outage according to the definition in Equation (30). Once the user channels are known, \(C_{sys}\) can be calculated. The challenge is to consider all possibilities of the sum sizes of requested content files. Let us consider \(U\) users each selecting a file from a list of \(N\) items; the feasible space of the users' request vector has an exponential size of \(N^{U}\). Although we could use the backtracking algorithm to enumerate all the possibilities [39], the computational complexity is too high. To this end, we further propose a fast method for outage estimation, as explained in Algorithm 3.  
```
1:Run the model \(n\) times, and get \(n*N*U\) recommended contents;
2:Calculate the frequency of the file size of the above contents, then the PDF of file size in each user's recommended list could be obtained;
3:Assume that all users are independent of each other. Then, the joint PDF of sum file size of \(U\) users is the \(U-1\) convolution of the above PDF;
4:With the joint PDF of the sum file size of \(U\) users, the joint CDF could be obtained. Then, the outage is \(1-CDF(C_{sys})\).
```  
**Algorithm 3** The algorithm for outage estimation.  
In Algorithm 3, we assume that all users are independent of each other. Hence, the outage can be evaluated using statistical methods.  
The accuracy of our outage approximation method is evaluated via Monte Carlo simulations. In each simulation, a user request profile and user channel gains are randomly generated. The exact value of outage probability is obtained by solving a large incident of the problem defined in (29) and calculating the empirical probability. The approximated outage is obtained according to Algorithm 3. Figure 2 compares the exact and approximated outage in different settings. We can see that the estimated outage makes a fairly good approximation to the exact outage curve.