# Mobility Management Scheme with Mobility Prediction in Wireless Communication Networks
## 5 Numerical Results  
To obtain numerical results, the following parameters were assumed:  
\[U=10,\,V=1,\,\lambda_{c}=\lambda_{in}+\lambda_{out}=0.5+0.5=1.0\]  
Incoming and outgoing calls were generated according to the Poison Process with mean \(1/\lambda_{in}\) and \(1/\lambda_{out}\), respectively. The time spent in a cell followed a gamma distribution with parameters 1 and \(1/\lambda_{in}\).  
Figure 5 shows the number of registrations for each UE with various call-to-mobility ratio (CMR) and \(q\) when \(N_{c}=7\) (urban case). From the numerical results, it was observed that the number of registrations decreased as the moving probability, \(q\), to the predicted direction increased, and the number of registrations decreased as CMR increased. Above all, it was observed that analytical and simulation results were very close. It could be concluded that the semi-Markov process model gave accurate results since differences between analytical results and simulation results were less than 1.42% on average. Therefore, subsequent numerical results were obtained using the semi-Markov process model. Therefore, subsequent mathematical results were obtained using the semi-Markov process model.  
Figure 5: Number of registrations versus call-to-mobility ratio (CMR).  
Figure 6 shows total costs for various CMR and \(q\) in urban case. From the figure, it was observed that the total cost decreased as CMR increased because of less cell residence time, and that the total cost decreased as \(q\) increased because of mobility prediction effect. It was also observed that, if CMR < 1 corresponding to current small cell configurations was considered, the registration scheme with our flexible TAL forming outperformed the distance-based registration (DBR) scheme with \(N_{c}\) = 7 when \(q\geq 0.75\), although the registration cost of the proposed scheme with random walk mobility (\(q\) = 1/6) was higher compared to the performance of the DBR.  
Figure 7 shows total costs for various CMR and \(q\) in rural case. Similar to urban case, from the figure, it was observed that, even though absolute value of the total cost is large compared to urban case, the total cost decreased as CMR increased, and the total cost decreased as \(q\) increased. It was also observed that, if CMR < 1 corresponding to current small cell configurations was considered, the registration scheme with our flexible TAL forming outperformed the original TAL scheme with \(N_{c}\) = 10 at \(q\geq 0.85\). From numerical results, we had the following observations:  
1. The number of registrations (or registration cost) decreased as the CMR increased because the UE resided longer in a cell with less registrations required.
2. The number of registrations decreased as the moving probability, \(q\), to the predicted direction increased. Note that, as \(q\) increased, the number of UEs moving to the center cell (the state S\({}_{2}\) in Figures 2 and 3) increased while the registration cost decreased.
3. If CMR < 1 corresponding to current small cell configurations was considered, the registration scheme with our flexible TAL forming outperformed the typical DBR and the original TAL-based scheme.  
Finally, Figure 8 shows performance improvement by implicit registration on total signaling cost. In general, when implicit registration is adopted, the registration cost is reduced. For example, when CMR = 1, if implicit registration is not adopted, the total cost would be 8.384 in the urban case. On the other hand, if implicit registration is adopted, the total cost would be 8.095. In the rural area, the total cost will be also reduced to 10.914  
Figure 6: Total cost versus CMR and \(q\) (urban case).  
compared to 11.151 when implicit registration is not adopted. From results shown in Figure 8, the total cost can be decreased by 2.16% if implicit registration is adopted.  
Figure 8: Performance improvement of implicit registration.  
Figure 7: Total cost versus CMR and \(q\) (rural case).