## 5 Performance Evaluation  
### Model Verification  
In this section, we present a series of simulation results to verify the correctness of the proposed analytical model in Figures 4-8. The simulation results are obtained by MATLAB. We focus on five performance indexes, i.e., average number of allocated preambles, average value of \(p_{ACB}\), average preamble utilization rate, average number of active devices and average access delay for DS devices as expressed by Equations (29)-(31), Equation (35) and Equation (37). It should be noted that in the actual scene, the value of \(M\) is far greater than \(L_{avail\_DS}\) since we are considering the simultaneous DS devices' access environment. However, in order to facilitate the model verification, we take a small value of \(M=30\). In addition, to fully explain the results, the value of \(L_{avail\_DS}\) is not set according to the actual configuration of LTE-A system. The specific parameters of model verification settings are listed in Table 1.  
From Figures 4-8, we can find that the curves of our analysis model are basically consistent with the simulation curves. The curves can well exhibit the influence of different parameters on the performance of the system in our proposed scheme. Figure 4 shows the effect of different parameters in terms of average access delay. We set two value of delay requirements, i.e., \(D_{req}=22\,\mathrm{ms}\) space between numbers and units and units not in italic and \(D_{req}=50\,\mathrm{ms}\). It can be seen clearly that the smaller the value of \(D_{req}\) is, the better the delay performance will be, this reflects the good performance in satisfying the delay requirement of our scheme. A larger value of \(M\) means a larger maximal number of active devices in each slot, thus it will lead to a greater peak delay. Average number of active devices varying different values of \(D_{req}\) and \(M\) is depicted in Figure 5. Through comparison of the results of \(M=50\) with the results of \(M=80\), we can find that if we set a larger number of \(M\), the average number of active devices will increase. Furthermore, with the increase of \(D_{req}\), the number of average active devices with a slot will become larger. Figures 6 and 7 respectively show the performance of average value of \(p_{ACB}\) and average number of allocated resources. In Figure 6, we can see that when we take a larger value of \(D_{req}\), the average value of ACB factor will become smaller. With the increase of the arrival rate of DS devices, \(p_{ACB}\) will decrease to a minimum value. As there will be more active devices when we take a larger value of \(M\), the minimum value of \(p_{ACB}\) for \(M=80\) is much smaller than the value for \(M=50\). The variation of average number of allocated preambles is shown in Figure 4. We can see that the curves for various \(T_{req}\) and \(M\) are relatively close to each other. Whereas the value for \(D_{req}=22\,\mathrm{ms}\) is slightly more than the value for \(D_{req}=50\,\mathrm{ms}\), representing that there are more RA resources reserved for DS devices when we have a more stringent delay requirement. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(M\) & 30 \\ \(L_{avail\_DS}\) & 50, 80 \\ \(T_{slot}\) & 10 ms \\ \(D_{req}\) & 22 ms, 50 ms \\ \(\lambda\) & 200–2000 arrivals/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter used in model verification. Average preamble utilization rate varying different values of \(D_{req}\) and \(M\) is depicted in Figure 8. Through comparison of the result of \(D_{req}=22\,\mathrm{ms}\) and the result of \(D_{req}=50\,\mathrm{ms}\), If we set a larger value of \(D_{req}\), the value of average preamble utilization rate will increase. The explanation for the results lies in the implementation process of the scheme that when the constraint of delay becomes looser, more preambles are allocated simply aiming at maximizing the preamble utilization rate with little attention to the access delay.