# Secrecy Control of Wireless Networks with Finite Encoding Blocklength
## 4 Numerical Results and Discussions  
For the network model presented in Figure 1, we consider i.i.d Rayleigh fading channels between nodes. The ratio of transmit power and noise has been normalized to 1. Let \(h_{i,j}\) be the power gain between node \(i\) and \(j\), which follows exponential distribution and the mean of each link is presented in Table 2. The achievable rate between node \(i\) and \(j\) is \(R_{i,j}(t)=\log(1+h_{i,j}(t))\) and the rate of eavesdropper \(R_{i,e}(t)=\log(1+h_{i,e}(t))\). The utility function is a logarithmic utility function, i.e.,\(U_{n}^{c}(t)=\kappa+\log(R_{n}^{cs}(t))\), where \(\kappa=3\) and \(R_{n}^{cs}(t)\) is the confidential rate selected by node \(n\) in slot \(t\). We assume the confidential data arrival process for each user follows an i.i.d Bernoulli process with rate \(\lambda\). Firstly, we analyze the performance of MSCAFB algorithm. In the simulation, the maximum average confidential data arrival rate is 2 bit/s, since the bandwidth is assumed to be one. Choosing the parameter \(V\in\{10,20,80\}\), we get the results in Figures 3 and 4, where each value is collected by running 5000 times. Figure 3 shows the impact of increasing the average confidential data arrival rate on the utility function and Figure 4 depicts the average queue backlog in the network. From the simulation results, we find: (1) For a fixed \(V\), when the arrival rate is low, Figure 3 indicates that the utility function linearly increases with the average admission confidential rate. The reason for this is that, if the arrival confidential rate is low, almost all the arrival confidential data can be admitted. (2) When the arrival confidential rate is larger than the secrecy channel capacity, the average admitted confidential rate turns into saturation. Not surprisingly, as the parameter \(V\) increases, we observe that the utility function grows closer to the optimal value. (3) While in Figure 4, the average queue backlog is increased with \(V\) dramatically. It indicates that the transmission delay is increased with \(V\). Thus, the choice of \(V\) is indeed a tradeoff between average utility and short-term system performance. To achieve both large utility and low delay, we will discuss the selection of parameter \(V\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline (\(S_{1}\),1) & (\(S_{1}\),2) & (\(S_{1}\),3) & (1,\(d_{1}\)) & (2,\(d_{1}\)) & (3,\(d_{1}\)) \\ \hline
6 & 8 & 10 & 8 & 6 & 4 \\ \hline (\(S_{2}\),2) & (\(S_{2}\),3) & (\(S_{2}\),4) & (2,\(d_{2}\)) & (3,\(d_{2}\)) & (4,\(d_{2}\)) \\ \hline
6 & 8 & 10 & 8 & 6 & 4 \\ \hline (\(S_{1}\),\(e\)) & (\(S_{2}\),\(e\)) & (1,\(e\)) & (2,\(e\)) & (3,\(e\)) & (4,\(e\)) \\ \hline
3 & 2 & 1 & 1 & 2 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean Channel Gain. Figure 3: Long-term utility with varying confidential data arrival rate. According to [22], the long-term average utility is proportion to \(1/V\), such that we can rewrite the utility function as \(\sum_{n,c}U_{n}^{c}(r_{n}^{cs}(t))\approx\sum_{n,c}U_{n}^{c}(r_{n}^{*cs})-BM/V\), where \(\sum_{n,c}U_{n}^{c}(r_{n}^{*cs})\) denotes the optimal value and is a constant. Hence, the utility function is an increasing hyperbolic function of parameter \(V\) and a good operating point would be to pick a \(V\) value where an unit increases in \(V\) yields a very small reduction in utility. At this point, the utility gains may not be worth the delay increase resulting from increasing \(V\) (since delay is proportional to \(V\)). Let \(\eta>0\) be the slope of utility function \(\sum_{n,c}U_{n}^{c}(r_{n}^{cs}(t))\). Differentiating to \(V\), i.e., \(d\left(\sum_{n,c}U_{n}^{c}(r_{n}^{*cs})-BM/V\right)/dV=BM/V^{2}=\eta\), we have the good operating point of \(V=\sqrt{BM}\), where \(M\) is the number of nodes, and \(B\triangleq\frac{1}{M}\sum_{n=1}^{M}\left[\left(\lambda_{n}^{cmax}+R_{n,max}^{ in,s}\right)^{2}+(R_{n,max}^{out,s})^{2}\right]\).