# A Petri Net Model for Cognitive Radio Internet of Things Networks Exploiting GSM Bands +
## 5 Numerical Results
When setting the primary packet switching load to \(N_{d}=1\), the distribution of the enqueued IUs in the secondary user appears to be almost linearly increasing for \(QS=5\) (Figure 7c,e, \(N_{d}=1\)), while it appears to be almost exponentially increasing for the others in terms of the considered queue sizes (Figures 7d,f and 8c-f, \(N_{d}=1\)). For higher values of the primary packet switching load (\(N_{d}=2\), \(N_{d}=3\), \(N_{d}=4\)), all of the distributions appear to be exponentially increasing. These conditions means an overload of the SU buffer with a high loss probability as the outcome. Figures 9 and 10 show the results in terms of the cumulative distribution function (CDF) of the response time. The end of the knee in these curves corresponds to the maximum response time. As in the previous figures, each subfigure shows the results for a specific primary circuit switching load \(Wl\) and specific queue size \(QS\). The primary packet switching load (\(N_{d}\)) is depicted by the respective curves in each subfigure, as indicated in the legends. As expected, the knees of the curves move to the right as the queue size increases. Taking into account a primary packet switching load with \(N_{d}>1\), it occurs at approximately 0.5 s for \(QS=5\) (Figure 9a,c,e, \(N_{d}>1\)), at 1.0 s for \(QS=10\) (Figure 9b,d,f, \(N_{d}>1\)), at 1.4 s for \(QS=15\) (Figure 10a,c,e, \(N_{d}>1\)), and at 1.8 s for \(QS=20\) (Figure 10b,d,f, \(N_{d}>1\)), regardless of the primary switching circuit load \(W_{l}\). For the lighter primary packet switching load (\(N_{d}=1\)), the knee of the curves moves slightly to the left; the moving appears to be higher for higher queue sizes. Figure 8: Performance results of the secondary user using a buffer size of \(15\) (**left column**) and \(20\) (**right column**) IUs. (**a**) \(Wl=0.3,QS=15\). (**b**) \(Wl=0.3,QS=20\). (**c**) \(Wl=1.0,QS=15\). (**d**) \(Wl=1.0,QS=20\). (**e**) \(Wl=3.0,QS=15\). (**f**) \(Wl=3.0,QS=20\). Figure 10: _Cont._  
Figure 9: Service time of the secondary user using a queue size of \(5\) (**left column**) and \(10\) (**right column**) IUs. (**a**) \(WI=0.3,QS=5\). (**b**) \(WI=0.3\), \(QS=10\). (**c**) \(WI=1.0,QS=5\). (**d**) \(WI=1.0\), \(QS=10\). (**e**) \(WI=3.0,QS=5\). (**f**) \(WI=3.0,QS=10\). We point out that the obtained CDFs are defective because the size of the queue is finite and a loss is experienced in almost all cases. The loss probability \(P_{l}\) is computed by taking into account the probability of having a full queue (i.e., \(P_{s}(5)\), \(P_{s}(10)\) of Figure 9 and \(P_{s}(15)\), \(P_{s}(20)\) of Figure 10); we can note that the loss probability corresponds to the value of the complement to 1 of each CDF at infinity (i.e., \(1-F_{r}(\infty)\)). Table 5 summarizes the value of loss probabilities in the different workload conditions that we analyzed when the SU application uses different buffer sizes, as modeled by resorting to different queue sizes \(QS\). Because the CDFs provide a complete characterization of the response time, it is easy to obtain other information about the SU behavior, such as the mean and variance of the delay to deliver IUs. With this detailed characterization of the response time, it is also possible to evaluate the \"average throughput\" (as the inverse of the mean value of the response time by the IU size), mean delay (the mean value of the response time), and jitter (as the variance of the response time) of the SU traffic and to characterize the system's QoS accordingly. All of these values are reported in Tables 6-8, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{3}{c}{\(\text{QS}=\text{5}\)} \\ \hline \hline \(W_{1}\)-\(Np\) & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) \\ \hline \(\mathbf{0.30}\) & 0.117 & 0.317 & 0.350 &