# On the Secrecy Sum-Rate of Internet of Things Networks: Scheduling and Power Control +
## 2 System Model
### Achievable Secrecy Sum-Rate  
It is difficult to accurately analyze an individual secrecy capacity region in wireless multiuser networks. Instead, we consider secrecy sum-rate as in [29]. Specially, we use a lower bound of the achievable secrecy sum-rate. For analytical simplicity, we assume that \(\alpha_{n}=1\) and \(\beta_{nk}=1\) for all \(n\) and \(k\). Then, the received signals at the base station, \(\mathbf{y}\in\mathbb{C}^{M\times 1}\), and at the \(k\)-th eavesdropper, \(\mathbf{y}_{k}\in\mathbb{C}^{M\times 1}\), are expressed, respectively, as  
\[\mathbf{y} =\sum_{s\in\mathcal{N}_{S}}\mathbf{h}_{s}x_{s}+\mathbf{z}, \tag{2}\] \[\mathbf{y}_{k} =\sum_{s\in\mathcal{N}_{S}}\mathbf{g}_{sk}x_{s}+\mathbf{z}_{k}, \tag{3}\]  
where \(x_{s}\) denotes the desired data symbol for the \(s\)-th transmitter among selected \(S\) transmitters (i.e., \(s\in\mathcal{N}_{S}\)), each of which meets the average power constraint \(P_{0}\), and \(\mathbf{z}\) and  
Figure 1: An uplink multiuser network consisting of \(N\) users (\(S\) scheduled transmitters for data), a single base station and \(K\) eavesdroppers: the SIMO MAC model.  
\(\mathbf{z}_{k}\) denote the independent and identically distributed and circularly symmetric complex additive white Gaussian noise vectors with zero mean and covariance of \(\sigma_{0}^{2}\mathbf{I}_{M}\) and \(\sigma_{e}^{2}\mathbf{I}_{M}\), respectively. For analytical tractability, we assume \(\sigma_{0}^{2}=\sigma_{e}^{2}\).  
Note that the achievable secrecy sum-rate is obtained based on the sum of secrecy rates between main channel (transmitters and base station) and wiretap channel (transmitters and eavesdroppers). Thus, the achievable secrecy sum-rate can be different depending on the post-processing at base station and eavesdroppers.  
For the main channel, the desired base station decodes received signal in Equation (2) by using receive random beamforming based on Equation (1) and, the achievable rate of each transmitter and base station pair in the main channel is given by  
\[r_{\text{BS}}^{[s]}=\log\left(1+\frac{|\mathbf{u}_{0}^{[s]\intercal}\mathbf{ h}_{\pi_{s}}|^{2}\rho}{1+\sum_{l=1,l\
eq s}^{S}|\mathbf{u}_{0}^{[l]\intercal} \mathbf{h}_{\pi_{l}}|^{2}\rho}\right), \tag{4}\]  
where \(\mathbf{u}_{0}^{[s]}\) represents \(s\)-th receive random beamforming vector defined in Equation (1). Here, subscript zero in \(\mathbf{u}_{0}^{[s]}\) indicates receive random beamforming vectors at the desired base station., \(\pi_{s}\) denotes scheduled transmitter index for the beam \(\mathbf{u}_{0}^{[s]}\), and \(\rho\) is the transmit SNR defined as \(\rho\triangleq\frac{P_{0}}{\sigma^{2}}\).  
Note that all signals except the desired signal of the scheduled transmitter for a beam index \(s\) in Equation (2) are considered interference and, thus, it is represented as denominators in the logarithm function in Equation (4).  
For the wiretap channel, we consider two eavesdropping scenarios: internal and external eavesdropping scenarios [32]. Depending on eavesdropping scenarios, the achievable secrecy sum-rate is differently defined, discussed in the next subsection.