# A Petri Net Model for Cognitive Radio Internet of Things Networks Exploiting GSM Bands +
## 5 Numerical Results  
In order to evaluate the proposed model, we firstly had to characterize the GSM/EDGE PUs' phone and data traffic. As specified in Section 4.1, according to [55], we set an exponential distribution of phone call length and packet data size. In order to consider different PU phone traffic loads, we set the average call holding time to 180 s in our experiments (\(\lambda_{s}=5.5556\times 10^{-3}\) s\({}^{-1}\)) and varied the mean rate of the Poisson process modeling call arrivals involving the specific transponder from 6 to 60 calls per hour (6, 20, and 60). Accordingly, we loaded each TCH with 0.043, 0.143, and 0.419 Erlang; as a consequence, the considered system workload \(Wl=\frac{\lambda_{d}}{\lambda_{s}}\) was set to 0.3, 1.0, and 3.0 Erlang. The mean duration of the on (talk spurt) period \(\frac{1}{\lambda_{s}}\) was set to be equal to 1 s, and the mean duration of the off (silent gap) period \(\frac{1}{\lambda_{sg}}\) was set to be equal to 1.35 s, which was the procedure for both in [55]. Regarding the packet switching data traffic, we considered the message length as exponentially distributed with an average length of 1.6 kB ([55]) and an average transmission rate equal to 34.2 kb/s. Accordingly, we set the expected rate of the packet switching data service as \(\lambda_{c}=2.61\) s\({}^{-1}\). In order to consider different PU data traffic loads, we varied the mean rate of the Poisson process modeling packet arrivals involving the specific transponder in our experiments. Specifically, while taking into account that a single user usually generates an average of 0. (30) packets/s ([55]), we varied the packet switching traffic workload considering \(N_{D}=1,2,3\), and 4 users (\(\lambda_{d}=0. (30)\), 0. (60), 0. (90), and 1. (21) s\({}^{-1}\)). We solved the PU model, deriving the probabilities \(P_{s-CS}\) and \(P_{s-CS+PS}\) finding a slot free for each time the SU wanted to use it at these different phone and data load conditions. The results are summarized in Tables 3 and 4. We used the WebSPN software tool [62] to numerically solve the NMSPN models. WebSPN also provides an analytical solution engine for models with many non-exponentially \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(W_{1}N_{D}\) & **1** & **2** & **3** & **4** \\ \hline
**0.3** & 0.993 & 0.993 & 0.993 & 0.993 \\
**1.0** & 0.977 & 0.977 & 0.977 & 0.977 \\
**3.0** & 0.933 & 0.933 & 0.933 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Probability to find a CS slot free (\(P_{s-CS}\)). Figure 6: Modification to the performance model by implementing the tagged customer technique. distributed transitions; thus, we did not need to resort to simulation for deriving the numerical results from the NMSPN models. The modeled SU node produces an \(N_{\text{B}}=8\) kbit IU with a mean rate of \(\lambda_{B}=16.64\) s\({}^{-1}\). We assumed a transmission rate of \(\lambda_{G}=271\) kb/s and, accordingly, the deterministic firing time of place End will be \(\tau_{UI}=0.03023\) s. Moreover, we assumed that the spectrum sensing algorithm needs 25% of a slot time to run, so we set \(f_{n}=0.25\) in the SU model and an accuracy such that \(P_{TP}=P_{TN}=0.95\). Figures 7 and 8 show the results obtained in terms of the probability distribution of \(n\) IUs in the queue. Specifically, we evaluated the probabilities changing the queue size \(QS\) in the set of values \(\{5,10,15,20\}\). Each subfigure shows the results for a specific primary circuit switching load \(Wl\) and a 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 distributions are affected by the primary loads for both circuit switching and packet switching. For the lighter packet switching and circuit switching loads (Figures 7a,b and 8a,b, \(N_{d}=1\)), the distributions appears to be almost uniform, regardless of the queue size.