# Mobility Management Scheme with Mobility Prediction in Wireless Communication Networks
## 3 Modeling and Performance Analysis
### Semi-Markov Process Model  
In both cases, the time spent in state S\({}_{0}\) is different from the time spent in other states. The time spent in state S\({}_{0}\) is the interval from the call occurrence to/from a UE in a cell to the time it moves to a neighboring cell or when a call occurs again. On the other hand, the time spent in other states except for state S\({}_{0}\) is the interval from the time the UE enters a cell to the time it moves to a neighboring cell or when a call occurs.  
To properly analyze the performance, different time spent in each state should be considered. First, the time spent in state S\({}_{0}\) can be expressed as follows:  
\[T_{c},\ if\ \ T_{c}\leq R_{m}R_{m},\ if\ \ T_{c}>R_{m} \tag{7}\]  
Its mean can be obtained as follows:  
\[\begin{array}{ll}\tau_{0}&=\int_{0}^{\infty}\int_{0}^{\tau_{m}}t_{c}f_{c}(t_ {c})dt_{c}f_{r}(r_{m})dr_{m}+\int_{0}^{\infty}\int_{\tau_{m}}^{\infty}r_{m}f_{ c}(t_{c})dt_{c}f_{r}(r_{m})dr_{m}\\ &=\int_{0}^{\infty}[\int_{0}^{\tau_{m}}t_{c}f_{c}(t_{c})dt_{c}+\int_{\tau_{m}} ^{\infty}r_{m}f_{c}(t_{c})dt_{c}]f_{r}(r_{m})dr_{m}=\frac{1}{\lambda_{c}} \left(1-\frac{\lambda_{m}}{\lambda_{c}}[1-f_{m}^{*}(\lambda_{c})]\right)\end{array} \tag{8}\]  
Next, the time spent in other states except for state S\({}_{0}\) can be expressed as follows:  
\[T_{c},\ \ \ \ \ if\ \ T_{c}\leq T_{m}T_{m},\ \ \ \ \ \ if\ \ T_{c}>T_{m} \tag{9}\]  
Its mean \(\tau_{i}\) (\(i\) = 1, 2,..., 5) can be obtained as follows:  
\[\begin{array}{ll}\tau_{i}&=\int_{0}^{\infty}\int_{0}^{t_{m}}t_{c}f_{c}(t_{c })dt_{c}f_{m}(t_{m})dt_{m}+\int_{0}^{\infty}\int_{t_{m}}^{\infty}t_{m}f_{c}(t_ {c})dt_{c}f_{m}(t_{m})dt_{m}\\ &=\int_{0}^{\infty}[\int_{0}^{t_{m}}t_{c}f_{c}(t_{c})dt_{c}+\int_{t_{m}}^{ \infty}t_{m}f_{c}(t_{c})dt_{c}]f_{m}(t_{m})=\frac{1}{\lambda_{c}}(1-f_{m}^{*} (\lambda_{c}))\end{array} \tag{10}\]  
To obtain the steady-state probability, \(\widetilde{\pi}\), considering different time spent in a state, we first calculated the steady-state probability \(\pi\) for the typical Markov chain [24] with transition probability matrix \(\mathbf{P}\) (\(\mathbf{P}_{urban}\) or \(\mathbf{P}_{runl}\)) using the following balanced equations:  
\[\pi P=\pi,\ \ \ \ \sum_{j}\pi_{j}=1 \tag{11}\]  
Finally, the steady-state probability of the semi-Markov process can be obtained as follows:  
\[\widetilde{\pi}_{i}=\frac{\pi_{i}\tau_{i}}{\sum_{j}\pi_{j}\tau_{j}} \tag{12}\]  
The total signaling cost on radio channels is the sum of the registration cost and the paging cost. The registration cost per UE per unit time, \(C_{U}\), can be obtained as follows:  
\[C_{U}=U\lambda_{m}\sum_{j\in B}[\widetilde{\pi}_{j}\cdotp_{U}(j)] \tag{13}\]  
where \(B\) is the set of states for boundary cells in each TAL and \(p_{U}(j)\) is the probability that a UE in state \(j\) enters a neighboring cell to request location registration.  
Assuming a simultaneous paging method, the paging cost per UE per unit time is as follows:  
\[C_{P}=V\lambda_{in}N_{c} \tag{14}\]  
where \(N_{c}\) is the number of cells in each TAL (\(N_{c}\) = 7 in urban, and \(N_{c}\) = 10 in rural). Finally, total cost can be obtained as follows:  
\[C_{T}=C_{U}+C_{P} \tag{15}\]