## 2 System Model
### Traffic Demand Model  
The asynchronous user behavior and the resulted traffic demand model are illustrated in Figure 2. The time is divided into frames of duration \(T\). Each time frame is further divided into \(n\) time slots. The list of content recommendation is updated at the beginning of each time frame. Within the time frame, we assume that users will browse the recommendation list and request content from the list in an asynchronous manner. It is assumed that the behavior for user \(u\) to request a content at each time slot follows a Bernoulli process with parameter \(\lambda_{u}\), which means the probability for user \(u\) to place a content request at the time slot \(t\) is  
\[P_{u,t}(x)=\left\{\begin{array}{ll}\lambda_{u},&x=1,\\ 1-\lambda_{u},&x=0.\end{array}\right. \tag{1}\]  
This model allows the user requests to be asynchronous in time and is a generalization of the model in [20]. When \(\lambda_{u}=0\), it means that user \(u\) is inactive and does not request any content at the time period. When \(\lambda_{u}=1\), it means that the user always requests one and only one piece of content at each time slot. In this case, our model reduces to the extreme case in [20]. The arrival rate \(\lambda_{u}\) is assumed to be stable during a time frame but can vary across different time frames and different users.  
Two general assumptions are further made about the traffic demand model. First, we assume that the size of a common content file (which can be a video) is much larger than the size of a recommendation list (which includes mainly text and small image), so that the transmission time of  
Figure 2: Illustration of the traffic demand model.  
the recommendation list is negligible. Second, we assume that the large scale (i.e., slow fading) channel gains remain unchanged during a cycle of content recommendation and delivery. This assumption is considered reasonable for indoor or pedestrian users.  
When a user generates a request according to the received recommendation list, we assumed that the probability of requesting any content in the recommended list is the same. When a specific piece of content is requested, traffic demand with respect to the content size is generated. As shown in Figure 2, different content sizes will result in varying traffic demands across different time slots. The probability mass function (PMF) of the traffic demand from user \(u\) at time slot \(t\) is given by  
\[Q_{u}^{t}(q)=\left\{\begin{array}{ll}1-\lambda_{u},&q=0,\\ \sum_{f\in L_{u}}\frac{\lambda_{u}}{N}\delta(q-L_{uf}),&\text{otherwise,}\end{array}\right. \tag{2}\]  
or expressed as  
\[Q_{u}^{t}(q)=(1-\lambda_{u})\delta(q)+\sum_{f\in L_{u}}\frac{\lambda_{u}}{N} \delta(q-L_{uf}), \tag{3}\]  
where \(N\) is the total number of files in the recommendation list, \(L_{u}\) is the set of indexes of the subset of contents included in the \(u\)-th user's recommendation list, and \(L_{uf}\) is the size of the corresponding file in the \(u\)-th user's recommendation list. Because we have a fixed number of files, the density of traffic demand at a time slot can only take discrete values of the file sizes.  
When there are multiple users, then PMFs of each user's traffic at time slot \(t\) are given by  
\[\begin{array}{l}Q_{1}^{t}(q)=(1-\lambda_{1})\delta(q)+\sum_{f\in L_{1}} \frac{\lambda_{1}}{N}\delta(q-L_{1f})\\ Q_{2}^{t}(q)=(1-\lambda_{2})\delta(q)+\sum_{f\in L_{2}}\frac{\lambda_{2}}{N} \delta(q-L_{2f})\\ \vdots\\ Q_{LU}^{t}(q)=(1-\lambda_{LU})\delta(q)+\sum_{f\in L_{U}}\frac{\lambda_{U}}{N }\delta(q-L_{Uf}).\end{array} \tag{4}\]  
The PMF of the sum traffic \(Q^{t}=q_{1}^{t}+q_{2}^{t}+\cdots+q_{LU}^{t}=\sum_{u=1}^{U}q_{u}^{t}\) at time slot \(t\) is then  
\[Q^{t}(q)=Q_{1}^{t}(q)\otimes Q_{2}^{t}(q)\otimes\cdots\otimes Q_{LU}^{t}(q), \tag{5}\]  
where \(\otimes\) represents the convolution operation.