## 5 Performance Evaluation
### Performance Analysis  
In this section, we report the validation results by comparing our proposed resource allocation and ACB scheme with the existing schemes for average access delay for DS devices, and average preamble utilization rate for DS devices and average number of preambles remained for DT devices. It should be noted that in order to simplify the demonstration of the performance of DT devices, we only considered the number of preambles remained for them, and the discussion of other performance indexes remains for our future work. As we have defined the total number of available preambles as \(L_{total}\), the average number of preambles allocated to DS devices as \(E[N_{L\_DS}]\) which has been described in Equation (29), we can obtain the average number of preambles remained for DT devices as:  
\[E[N_{L\_DT}]=L_{total}-E[N_{L\_DS}] \tag{38}\]  
A typical configuration of the total number of available preambles in an RA slot is \(L_{total}=64\)[25]. In order to guarantee the communication requirements of DS devices, we set \(L_{avail\_DS}\) to a relatively large value 54. Considering that in actual system, the number of active devices within a slot can be much greater than the amount of available RA resources, we set \(M\) as 500. Other detailed parameters are set based on reference [25] and they are listed in Table 2.  
With respect to the comparison targets, we consider the following conventional methods which are respectively expressed by scheme A, scheme B, scheme C and scheme D: A. fixed preamble allocation with fixed value of \(p_{ACB}\) as described in [30]; B. fixed preamble allocation with dynamic tuning of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Descriptions** & **Notation** & **Value** \\ \hline Maximum number of active devices to be handled in a slot & \(M\) & 500 \\ Arrival rate of DS devices & \(\lambda\) & 300–3000 arrivals/s \\ Total number of available preambles in an RA slot & \(L_{total}\) & 64 \\ Maximum number of available preambles for DS Devices in an RA slot & \(L_{avail\_DS}\) & 54 \\ Length of a random access slot & \(T_{slot}\) & 10 ms \\ Delay requirement of DS devices & \(D_{req}\) & 22 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used in performance analysis. Figure 8: Average Preamble Utilization Rate with different \(L_{avail\_DS}\) and \(D_{req}\). \(p_{ACB}\) as described in [33]; C. dynamic preamble allocation with fixed value of \(p_{ACB}\) and D. dynamic preamble allocation with dynamic tuning of \(p_{ACB}\) as described in [18]. In [30; 33], the method for estimating the number of active devices is not discussed. In [18], load estimation is based on the preamble collision rate in previous slots. In order to facilitate performance comparison, we assume eNodeB has perfect knowledge of number of active devices for these schemes. In scheme A, we set \(L_{DS}\) as its maximum value 54, \(p_{ACB}\) as 0.2. In scheme B, the fixed number of preambles is set as 54, eNodeB will dynamically adjust the value of \(p_{ACB}\) to \(L_{avail\_DS}/N_{pa}\) to maximize the preamble utilization rate. In scheme C, \(p_{ACB}\) is set as 0.2, eNodeB adjust the value of \(L_{DS}\) using the similar mechanism as scheme B. In the implementation of scheme D, we take the value of parameter \(b\) in [18] as 1 and dynamically adjust \(L_{DS}\) and \(p_{ACB}\). Figures 9 and 10 show the performances of average access delay and average preamble utilization rate for DS devices, respectively. In order to facilitate our comparison, we drew a straight line with the delay of 22 ms in Figure 9. We can see from the result that among these five schemes, the scheme of fixed preamble allocation with dynamic \(p_{ACB}\) shows better delay performances, it is because in this scheme, there are sufficient preamble resources reserved for DS devices. This scheme has the ability to satisfy the delay requirement in the wide range of the operating region.