# On the Secrecy Sum-Rate of Internet of Things Networks: Scheduling and Power Control +
## 4 Secrecy Sum-Rate Scaling Analysis
Then, \(\lim\limits_{x\to\infty}(1-f(x))^{x}=0\) if and only if \(\lim\limits_{x\to\infty}xf(x)\to\infty\)._  
Proof: If \(\lim\limits_{x\to\infty}xf(x)\to\infty\), then it follows that \(f(x)=\omega(\frac{1}{x})\)[31], thus resulting in  
\[\lim\limits_{x\to\infty}(1-f(x))^{x}=o\left(\lim\limits_{x\to\infty}\left(1- \frac{1}{x}\right)^{x}\right)=o(1)\]  
for \(0<f(x)\leq 1\). It is hence seen that \(\lim\limits_{x\to\infty}(1-f(x))^{x}\) converges to zero. If \(\lim\limits_{x\to\infty}xf(x)\) is finite, then there exists a constant \(c_{3}>0\) such that \(xf(x)<c_{3}\) for any \(x\geq 0\). We then have  
\[\lim\limits_{x\to\infty}(1-f(x))^{x}=\lim\limits_{x\to\infty}\left(1-\frac{c_ {3}}{x}\right)^{x}=e^{-c_{3}}>0,\]  
which completes the proof. Let \(p^{[l^{*}]}\) denote a probability that at least one transmitter satisfying all criteria in (13) for \(l^{*}\)-th beam. To analyze \(p^{[l^{*}]}\), we characterize the probability that each criterion is satisfied for a certain transmitter (i.e., \(\Pr(\mathbf{C1})\), \(\Pr(\mathbf{C2})\), and \(\Pr(\mathbf{C3})\)). Hereafter, we omit the transmitter index \(n\) and the beam index \(l^{*}\) for representing each probability since we assume i.i.d. channel vectors. In other words, \(\Pr(\mathbf{C1})\), \(\Pr(\mathbf{C2})\), or \(\Pr(\mathbf{C3})\) are the same regardless of the transmitter index \(n\in\{1,\cdots,N\}\) and beam index \(l^{*}\in\{1,\cdots,M\}\). First, \(\Pr(\mathbf{C1})\) is given by  
\[\Pr(\mathbf{C1})\triangleq\Pr\left\{|\mathbf{u}_{0}^{[l^{*}]T}\mathbf{h}_{n}| ^{2}\geq\eta_{\mathrm{Q}}^{*}\right\}=e^{-\eta_{\mathrm{Q}}^{*}}, \tag{14}\]  
since the receive beam \(\mathbf{u}_{0}^{[l^{*}]T}\) is assumed to be isotropically distributed and thus \(|\mathbf{u}_{0}^{[l^{*}]T}\mathbf{h}_{n}|^{2}\) is exponentially distributed [28]. Second, \(\Pr(\mathbf{C2})\) is given by  
\[\Pr(\mathbf{C2}) \triangleq\Pr\left\{\sum\limits_{l=1,l\
eq l^{*}}^{S}|\mathbf{u}_ {0}^{[l]T}\mathbf{h}_{n}|^{2}\leq\eta_{1}^{*}\right\}\] \[=\frac{\gamma(S-1,\eta_{1}^{*}/2)}{\Gamma(S-1)}, \tag{15}\]  
where \(\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt\) is the Gamma function and \(\gamma(z,x)=\int_{0}^{x}t^{z-1}e^{-t}dt\) is the lower incomplete Gamma function. In (15), the last equality holds due to the fact that \(|\mathbf{u}_{0}^{[l]T}\mathbf{h}_{n}|^{2}\) is exponentially distributed and the sum of \(S-1\) independent exponential random variables is distributed according to the chi-square distribution with \(2(S-1)\) degrees-of-freedom [35]. Third, \(\Pr(\mathbf{C3})\) is given by  
\[\Pr(\mathbf{C3}) \triangleq\Pr\biggl{\{}\max_{k\in\mathcal{K}}\lVert\mathbf{g}_{nk} \rVert^{2}\leq\eta_{\mathrm{L}}^{\star}\biggr{\}}\] \[=\left(\frac{\gamma(M,\eta_{\mathrm{L}}^{\star}/2)}{\Gamma(M)} \right)^{K}, \tag{16}\]  
since the term \(\lVert\mathbf{g}_{nk}\rVert^{2}\) is the sum of \(M\) independent exponential random variables and thus it is distributed according to the chi-square distribution with \(2M\) degrees-of-freedom (i.e., \(\frac{\gamma(M,\eta_{\mathrm{L}}^{\star}/2)}{\Gamma(M)}\)). Therefore, according to [35], the term \(\max_{k\in\mathcal{K}}\lVert\mathbf{g}_{nk}\rVert^{2}\) is distributed as \(\left(\frac{\gamma(M,\eta_{\mathrm{L}}^{\star}/2)}{\Gamma(M)}\right)^{K}\). Additionally, \(\Pr(\mathbf{C2})\) and \(\Pr(\mathbf{C3})\) can be lower-bounded by using the following lemma. **Lemma 2**. : _For any \(0\leq x<1\) and \(z>0\), the lower incomplete Gamma function \(\gamma(z,x)\) is lower-bounded by_  
\[\gamma(z,x)\geq\frac{1}{z}x^{z}e^{-1}. \tag{17}\]  
Proof. : The inequality in (17) holds since  
\[\gamma(z,x) =\frac{1}{z}x^{z}e^{-x}+\gamma(z+1,x)\] \[=\frac{1}{z}x^{z}e^{-x}+\frac{1}{z(z)}x^{z+1}e^{-x}+\cdots\] \[\lim\limits_{x\to\infty}(1-f(x))^{x}=\lim\limits_{x\to\infty}\left(1-\frac{c_ {3}}{x}\right)^{x}=e^{-c_{3}}>0,\]0  
which completes the proof. Finally, we introduce our main theorem to show the achievable secrecy sum-rate scaling of the proposed scheduling scheme. **Theorem 1**.