# Analyzing Zone-Based Registration Using a Three Zone System: A Semi-Markov Process Approach
## 3 New Mathematical Model
### Semi-Markov Process Model for 3Z
#### 3.2.6 Calculation of Sojourn Time in Each State  
It is necessary to note that the sojourn time of state I\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) is different from the sojourn time of state E\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) or state C\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3). The sojourn time of state E\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) or state C\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) is the interval from the time a UE enters a zone until the time it leaves. On the other hand, the sojourn time of state I\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) is the interval from the time a call to/from a UE occurs in a zone, until the time it leaves.  
In the transition diagram, the UE in state E\({}_{21}\) (the state that the UE registered when moving from the center zone to an edge zone) can transit to one of three states: (i) If it moves to a new zone (with probability \((1-\theta)P(T_{c}>T_{m})=(1-\theta)m\)), its state becomes E\({}_{21}\) again. (ii) If it returns to the last zone (with probability \(\theta P(T_{c}>T_{m})=\theta m\)), its state becomes E\({}_{12}\). (iii) If a call from/to the UE occurs before the UE moves to a neighboring zone (with probability \(P(T_{c}<T_{m})=1-P(T_{c}>T_{m})=1-m\)), UE's state changes to state I\({}_{21}\).  
We should consider the sojourn time of each state to be different, in order to derive an accurate probability of the first paging failure.  
Now, let us derive the sojourn time of state I\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3). Since the sojourn time of state I\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) can be expressed as:  
\[T_{I}\)=\begin{cases}T_{c},&if\ T_{c}\leq R_{m}\\ R_{m},&if\ T_{c}>R_{m}\end{cases}\]  
and its mean can be derived as follows:\[\begin{array}{ll}\tau_{I}&=\int_{0}^{\infty}\int_{0}^{\tau_{m}}t_{c}f_{c}(t_{c}) dt_{c}f_{r}(r_{m})dr_{m}+\int_{0}^{\infty}\int_{\tau_{m}}^{\infty}r_{m}f_{c}(t_{c})dt_{c}f_{r}(r_{ m})dr_{m}\\ &=\int_{0}^{\infty}\left[\int_{0}^{\tau_{m}}t_{c}f_{c}(t_{c})dt_{c}+\int_{\tau_ {m}}^{\infty}r_{m}f_{c}(t_{c})dt_{c}\right]f_{r}(r_{m})dr_{m}\\ &=\int_{0}^{\infty}\left[\frac{1}{\lambda}\big{(}1-e^{-\lambda t_{m}}\big{)} \right]f_{r}(r_{m})dr_{m}=\frac{1}{\lambda}(1-f_{r}^{*}(\lambda))\\ &=\frac{1}{\lambda}(1-\frac{\mu}{\lambda}[1-f_{m}^{*}(\lambda)])\end{array} \tag{5}\]  
Next, let us derive the sojourn time of state E\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) or state C\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3). Since the sojourn time of state E\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) or state C\({}_{ij}\) (\(i\) = 1, 2, 3; \(j\) = 1, 2, 3) can be expressed as:  
\[T_{E}=\begin{cases}T_{c},&if\ T_{c}\leq T_{m}\\ T_{m},&if\ T_{c}>T_{m}\end{cases}\]  
and its mean can be derived as follows:  
\[\begin{array}{l}\tau_{E}=\tau_{C}=\int_{0}^{\infty}\int_{0}^{t_{m}}t_{c}f_{c} (t_{c})dt_{c}f_{m}(t_{m})dt_{m}+\int_{0}^{\infty}\int_{\tau_{m}}^{\infty}t_{m} f_{c}(t_{c})dt_{c}f_{m}(t_{m})dt_{m}=\\ \int_{0}^{\infty}[\int_{0}^{\tau_{m}}t_{c}f_{c}(t_{c})dt_{c}+\int_{\tau_{m}}^{ \infty}t_{m}f_{c}(t_{c})dt_{c}]f_{m}(t_{m})dt_{m}=\int_{0}^{\infty}[\frac{1}{ \lambda}\big{(}1-e^{-\lambda t_{m}}\big{)}]f_{m}(t_{m})dt_{m}=\\ \frac{1}{\lambda}(1-f_{m}^{*}(\lambda))\end{array} \tag{6}\]