## 4 Numerical Results  
In consideration of a realistic operational environment, the following parameters are used to compare the performances of the 1Z, 2Z, and 3Z systems:  
\[U=10,\,V=1,\,n=9,\,\mu=1,\,\lambda=\lambda_{o}+\lambda_{i}=0.5+0.5=1.0.\]  
When \(\mu=1\), the time residing in each zone follows an exponential distribution with an average of 1. Table 1 and Figure 10 show the results for the three systems (1Z, 2Z, 3Z) while assuming \(\theta=1/6\). As presented in Table 1, as CMR increases, that is, as the time spent in the zone increases, the registration cost decreases, after which the total cost decreases. The results for hexagonal zones showed that the 3Z outperformed the 1Z and 2Z and that the cost of 3Z was always less than the cost of 1Z. The results also showed that if the UEstayed in the zone for a long time, that is, if the CMR was large, the improvement was reduced compared to 1Z or 2Z. However, the performance of 3Z was still the best. Figure 10 shows the total cost and the relative cost reduction ratio. The left part of the vertical axis in Figure 10 represents the total cost, and the right part indicates the cost reduction ratio (_red. ratio_). The _red. ratio_ refers to the 2Z cost reduction ratio and the 3Z cost reduction ratio to the total cost of 1Z, respectively. Figure 10 implies that as the CMR decreased (as the zone residence time decreased), the cost reduction ratio of the 3Z (or 2Z) system increased compared to that of the 1Z system. Table 1 also shows the results for square-shaped zones (1Z_4, 2Z_4, 3Z_4) while assuming \(\theta=1/4\). The results showed that, in general, 3Z was better than 2Z in zone-based registration with hexagonal zones, and 2Z was better than 3Z in zone-based registration with square-shaped zones. Then, which performance was better, 3Z or 2Z_4? As shown in Table 1, in the environment assumed in this paper, it could be seen that 2Z_4 was slightly superior for CMR = 0.25 and CMR = 0.5, while 3Z was slightly superior for CMR = 1, 2, and 4. In conclusion, it can be seen that the hexagonal zone might be superior, or the square-shaped zone might be superior depending on the network environment. Note that the difference was not significant. The analytical results for CMR \(<1\) are shown in Figure 11. In the figure, the 2Z effect \(=(1Z-2Z)/1Z\times 100(\%)\), the 3Z effect \(=(1Z-3Z)/1Z\times 100(\%)\), and the _red. ratio_ of 3Z to 2Z are separately indicated. We could see that if CMR \(<1\), the cost at 3Z was always smaller than the cost at 1Z or 2Z. The smaller the CMR, the higher the reduction ratio. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**CMR (\(\lambda\)/\(\mu\))** & **1Z** & **2Z** & **3Z** & **1Z_4** & **2Z_4** & **3Z_4** & **(1Z \(-\) 2Z)/1Z (\%)** & **(1Z \(-\) 3Z)/1Z (\%)** \\ \hline
0.25 & 49.00 & 43.23 & 40.88 & 49.00 & 40.50 & 40.55 & 11.8 & 16.6 \\ \hline
0.5 & 29.00 & 26.36 & 25.42 & 29.00 & 25.29 & 25.96 & 9.1 & 12.3 \\ \hline
1 & 19.00 & 17.81 & 17.46 & 19.00 & 17.50 & 18.23 & 6.3 & 8.1 \\ \hline
2 & 14.00 & 13.46 & 13.33 & 14.00 & 13.44 & 13.99 & 3.9 & 4.8 \\ \hline
4 & 11.50 & 11.37 & 11.20 & 11.50 & 11.30 & 11.65 & 1.1 & 2.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Total cost for 1Z, 2Z, and 3Z systems (\(\theta=\frac{1}{6}\) for hexagonal zones, \(\theta=\frac{1}{4}\) for squared zones). Figure 10: Total cost and reduction ratio. Figure 12 compares the total costs for various values of \(\theta\). As a natural result, the 1Z system incurs costs regardless of \(\theta\). The results revealed that in the 2Z or 3Z systems, as \(\theta\) increased (that is, as the probability of returning to previous zone increased), the registration cost decreased, after which the total cost decreased linearly. Figures 13 and 14 show the registration and paging costs constituting the total cost as a function of \(\mu\) (the number of times entering the zone during unit time), respectively. As shown in Figure 13, as \(\mu\) increased, the residence time in the zone decreased, and the registration cost therefore increased linearly.