# Joint Content Recommendation and Delivery in Mobile Wireless Networks with Outage Management
## 2 System Model
### Wireless Transmission Model  
An orthogonal frequency division multiplexing (OFDM)-based multi-user wireless transmission system is assumed. The number of OFDM subcarriers is denoted as \(K\). Let us denote \(\alpha_{uk}^{2}\) as the instantaneous channel gain between the BS and the \(u\)-th user on subcarrier \(k\) (\(k=1,2,\ldots,K\)), \(c_{uk}\) as the number of bits allocated to user \(u\) on subcarrier \(k\) and \(p_{uk}\) as the transmit energy assigned to user \(u\) on subcarrier \(k\). We have \(p_{uk}=f(c_{uk})/\alpha_{uk}^{2}\), where \(f(c)\) represents the transmit energy required for  
\begin{table}
\begin{tabular}{l l l} \hline \hline  & **Pros** & **Cons** \\ \hline \multirow{3}{*}{Proactive caching protocol} & \multirow{3}{*}{1. Possibly less time delay.} & 1. Require large cache space; \\  & & 2. Low resource efficiency (due to invalid \\  & & transmission); \\  & & 3. Difficult traffic pricing, \\ \hline \multirow{3}{*}{Conventional pull-type protocol} & 1. High resource efficiency; & 1. Uncontrollable content access delay; \\  & 2. Small cache space and easy traffic pricing; & 2. Uncontrollable congestion probability. \\  & 3. High user interest. & 1. User interest may be compromised. \\ \hline \multirow{3}{*}{Proposed content retrieval protocol} & 1. High resource efficiency; & 1. User interest may be compromised. \\  & 2. Small cache space and easy pricing; & 1. User interest may be compromised. \\ \cline{1-1}  & 3. Controllable content access delay; & 1. User interest may be compromised. \\ \cline{1-1}  & 4. Controllable congestion probability. & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Pros and cons of the three types of protocols.  
the subcarrier to reliably receive \(c\) bits per symbol when the channel gain is one. It is assumed that the channel gains follow an exponential distribution [27; 28; 29]. The total transmit power of the BS is denoted as \(P_{T}\).  
We consider a \(M\) quadrature amplitude modulation (\(M\)-QAM) system. It follows that \(M=2^{c}\), where \(c\) is the number of bits carried by a QAM symbol. In this case, the bit error rate (BER) of the system is given by [30]:  
\[P_{e}\approx 4Q\left(\sqrt{\frac{d^{2}}{2N_{0}}}\right) \tag{1}\]  
where \(d\) is the minimum distance between two points in the signal constellation, \(N_{0}\) is the additive white Gaussian noise (AWGN) power spectral density and \(Q(\cdot)\) is the Q-function [31]. The average carrying energy of an \(M\)-QAM signal is [30]:  
\[f(c)=(2^{c}-1)d^{2}/6. \tag{2}\]  
Substituting Equation (1) into (2), we get [30]:  
\[f(c)=\frac{N_{0}}{3}\left[Q^{-1}\left(\frac{P_{e}}{4}\right)\right]^{2}(2^{c} -1). \tag{3}\]  
This equation establishes the required received power \(f(c)\) as a function of bits per symbol \(c\) at a target BER \(P_{e}\). Let us further define a binary variable \(\rho_{uk}\in\{0,1\}\) to denote whether channel \(k\) is assigned to user \(u\). When the channel is assigned, we have \(\rho_{uk}=1\). The energy assigned to user \(u\) is:  
\[P_{u}=\sum_{k=1}^{K}p_{uk}*\rho_{uk}=\sum_{k=1}^{K}\frac{f(c_{uk})}{\alpha_{uk} ^{2}}*\rho_{uk} \tag{4}\]  
where \(P_{u}\) is the energy required for user \(u\) to transmit \(\sum_{k=1}^{K}c_{uk}*\rho_{uk}\) bits per symbol. Therefore, the total power allocated to user \(u\) is:  
\[P_{u}^{total}=B*\sum_{k=1}^{K}p_{uk}*\rho_{uk} \tag{5}\]  
where \(B\) is the system bandwidth. Because the total power of the BS is constrained by \(P_{T}\), we have:  
\[B*\sum_{u=1}^{U}\sum_{k=1}^{K}p_{uk}*\rho_{uk}\leq P_{T}. \tag{6}\]  
Let us define \(S_{u}\) as the transmit data rate assigned to user \(u\), which is the sum of the number of bits in all subcarriers allocated to the user. It follows that:  
\[S_{u}=\sum_{k=1}^{K}c_{uk}*\rho_{uk}\qquad\text{(bit/symbol)}. \tag{7}\]  
Given the system bandwidth \(B\), the bit rate of user \(u\) is approximately:  
\[R_{u}=S_{u}*B=\sum_{k=1}^{K}c_{uk}*\rho_{uk}*B\qquad\text{(bit/s)}. \tag{8}\]