## 3 Modeling and Performance Analysis
### Sojourn Time and Exact Steady-State Probability  
According to the semi-Markov process approach [26], the steady-state probability of the UE should be recalculated using \(\mathbf{\pi}\) and the sojourn time spent in each state. First, for Cases 1 and 3, the sojourn time \(T_{case1}\) and \(T_{case3}\) of remaining in each state are expressed as follows.  
\[T_{case1}=T_{case3}=\left\{\begin{array}{ll}T_{c},&if\ T_{c}\leq T_{m}\\ T_{m},&if\ T_{c}>T_{m}\end{array}\right. \tag{6}\]  
Thus, we have the average sojourn time as follows:  
\[E(T_{case1})=E(T_{case3})=\int\limits_{\begin{array}{ll}\int\limits_{ \begin{array}{ll}t_{c}f_{c}(t_{c})f_{m}(t_{m})dt_{c}dt_{m}+\int\limits_{ \begin{array}{ll}\infty\,\infty\\ 0\,t_{m}\end{array}}t_{m}f_{c}(t_{c})f_{m}(t_{m})dt_{c}dt_{m}=\frac{1}{\lambda }(1-f_{m}^{*}(\lambda))\end{array}} \tag{7}\]  
Note that if a call occurs while the UE stays in the zone, the sojourn time is \(T_{c}\). If the call does not occur, the sojourn time is \(T_{m}\). Unlike Case 1 or 3, the sojourn time under Case 2 with the implicit registration connection is presented as follows:  
\[T_{case2}=\left\{\begin{array}{ll}T_{c},&if\ T_{c}\leq R_{m}\\ R_{m},&if\ T_{c}>R_{m}\end{array}\right. \tag{8}\]  
That is, when a new call happens again before entering another zone after implicit registration, \(T_{case2}=T_{c}\). If no call happens while entering another zone, it becomes the remaining residence time (\(T_{case2}=R_{m}\)) in the zone. The mean sojourn time for Case 2 becomes as follows:  
\[E(T_{case2})=\int\limits_{\begin{array}{ll}\int\limits_{ \begin{array}{ll}t_{c}f_{c}(t_{c})f_{r}(r_{m})dt_{c}dt_{m}+\int\limits_{ \begin{array}{ll}0\,\infty\\ 0\,\,0\end{array}}r_{m}\end{array}}r_{m}f_{c}(t_{c})f_{r}(r_{m})dt_{c}dt_{m}= \frac{1}{\lambda}(1-\frac{\mu}{\lambda}\overset{\ast}{f}_{m}^{*}(\lambda)) \tag{9}\]  
Therefore, the product \(\Phi\) of the steady-state probability and the sojourn time can be defined as follows:  
\[\Phi=E(T_{case1})\times\left\{\sum\
olimits_{i}\sum\
olimits_{j}\left(\pi_{E_ {ij}^{I}}+\pi_{E_{ij}^{II}}\right)\right\}+E(T_{case2})\times\left\{\pi_{I_{21 }^{I}}+\pi_{I_{12}^{I}}+\pi_{I_{21}^{II}}\right\}+E(T_{case3})\times\left\{\sum \
olimits_{i}\sum\
olimits_{j}\pi_{C_{ij}^{I}}\right\} \tag{10}\]  
Here, we can obtain the steady-state probability \(\mathbf{\pi}\) from the following balance equations and normalization conditions using the usual Markov chain [26]:  
\[\mathbf{P}=\mathbf{\pi},\ \ \ \sum\
olimits_{for\ all\ states}\pi=1 \tag{11}\]  
Finally, the exact steady-state probability \(\overset{\sim}{\mathbf{\pi}}\) can be obtained as follows:  
For Type I,\[\widetilde{\pi_{E_{ij}^{I}}}=\frac{E(T_{casel})\times\pi_{E_{ij}^{I}}}{\Phi}, \widetilde{\pi_{I_{21}^{I}}}=\frac{E(T_{casel})\times\pi_{I_{21}^{I}}}{\Phi}, \widetilde{\pi_{I_{22}^{I}}}=\frac{E(T_{casel})\times\pi_{I_{22}^{I}}}{\Phi}, \text{ and }\widetilde{\pi_{C_{ij}^{I}}}=\frac{E(T_{casel})\times\pi_{C_{ij}^{I}}}{\Phi} \tag{12}\]  
For Type II,  
\[\widetilde{\pi_{E_{ij}^{II}}}=\frac{E(T_{casel})\times\pi_{E_{ij}^{II}}}{\Phi},\text{ and }\widetilde{\pi_{I_{21}^{II}}}=\frac{E(T_{casel})\times\pi_{I_{21}^{II}}}{\Phi} \tag{13}\]