# Analyzing Zone-Based Registration under 2-Step Paging in Mobile Communication Network
## 3 Analysis of ZBR under 2-Step Paging
### Markov Chain Model and Analysis  
When the total number of cells in the zone is 4 (\(n=4\)), the movement of the UE can be modeled as shown in Figure 7 by using the Markov Chain (MC). State \(i\) indicates that the UE is in state \(i\).  
Figure 6: Inter-cell mobility model in a zone in 1 ZR when a UE moves from the zone in the left (\(n=9\)).  
Figure 7: State transition diagram and state transition matrix (\(n=4\)).  
Each element \(p_{ij}\) of the state transition matrix \(\mathbf{P}\) corresponding to the state transition diagram represents the probability of a transition from state \(i\) to state j by one cell entrance. For example, if the UE registers its zone and remains in state 1, it can move to the right cell within the zone with probability \(r\) and becomes state 2. Otherwise, it can move to the cell of the zone visited immediately before (with probability \(q\)), enter the upper cell (with probability \(r\)), or enter the cell below (with probability \(r\)), and remain in state 1.  
If the steady-state probability of state \(i\) in 1 ZR is \(\pi_{i}\), then \(\pi_{i}\) can be calculated using the following balance equations [20].  
\[\boldsymbol{\pi}=\boldsymbol{\pi}\mathbf{P},\;\sum_{i}\pi_{i}=1\;(i=1,\;2,\; 3,\ldots) \tag{1}\]  
For example, when \(n=4\), the steady-state probability can be calculated using the following equations.  
\[\begin{bmatrix}\pi_{1}&\pi_{2}\end{bmatrix}=\begin{bmatrix}\pi_{1}&\pi_{2} \end{bmatrix}\begin{bmatrix}q+2r&r\\ q+2r&r\end{bmatrix},\;\pi_{1}+\pi_{2}=1 \tag{2}\]  
When the total number of cells is 9 (\(n=9\)), six states are defined, as shown in Figure 8. In the case of \(n=9\), the state transition diagram and the state transition matrix can be obtained in a similar manner. For example, if the UE registers its zone and remains in state 1, it can move to the right cell within the zone with probability \(r\) and becomes state 3. Otherwise, it can move to the cell of the zone visited immediately before (with probability \(q\) and becomes state 1), enter the upper cell (with probability \(r\) and becomes state 1), or enter the cell below (with probability \(r\) and becomes state 2), and remain in state 1. Therefore, the transition probability from state 1 to state 2 (\(p_{12}\)) or state 3 (\(p_{13}\)) is \(r\), and the transition probability from state 1 to state 1 (\(p_{11}\)) is \(q+r\) since if the UE moves to the cells of neighboring zones it becomes state 1. Finally, the steady-state probability can be calculated using the balance equation.  
In the case of \(n=4^{2}=16\), \(n=5^{2}=25\),..., the state transition diagram and state transition probability matrix can be obtained similarly.  
Through this process, the probability that the UE is in a specific state in the zone can be obtained, and this can be used to determine how we divide and sequentially page n cells to obtain the minimal paging cost.  
Using the analysis results of 1 ZR under 2-step paging, let us compare the performance with 2 ZR. To apply 2-step paging in 1 ZR, a zone composed of several cells should be considered; however, in the previous study, only the movement between the zones was  
Figure 8: State transition diagram and state transition probability matrix (\(n=9\)).  
considered. Therefore, to compare the performance with the previous 2 ZR that only considered the time spent in the zone, it is also necessary to determine the time spent in the zone in the 1 ZR.