Analyzing Zone-Based Registration under 2-Step Paging in Mobile Communication Network

Abstract

In this study, we have considered zone-based registration (ZBR), which is the most widely adopted type in mobile communication networks. Based on a performance comparison between one-zone-based registration (1ZR) and two-zone-based registration (2ZR), 2ZR is known to be superior to 1ZR in most cases. However, the existing studies on the comparison of 1ZR and 2ZR have a critical problem: The basic assumption in 1ZR is that the entire zone is paged simultaneously when a call arrives, whereas, in 2ZR, the zone is paged by two steps. With these different paging schemes, a proper comparison cannot be made. Therefore, in this study, we analyzed the performance of 1ZR by adopting 2-step paging (2SP) such as 2ZR for a proper performance comparison under equivalent conditions. This study also presents an analytical model of assigning paging areas under 2SP in 1ZR when the zone consists of multiple cells. Considering the mobility characteristics assumed in the previous studies on 2ZR, we have presented the mobility model for movement among cells in a zone, obtained the steady-state probability of each cell by using the Markov chain model to calculate the paging cost under 2SP, and ultimately calculated the total signaling cost. Through various numerical results, it was observed that, when 1ZR also adopts 2SP such as 2ZR, 1ZR can be superior to 2ZR in many cases. In conclusion, by adopting 2SP in both 1ZR and 2ZR, it would be possible to reduce the total signaling cost by selecting the better 1ZR and 2ZR while considering traffic changes.

1. Introduction

To successfully track the location of user equipment (UE) in a 5G mobile communication network, it is necessary to use location registration and paging procedures. Location registration refers to a series of processes that involve continuously registering the location information in a network database to track the current location of a UE because the UE continues to move. When there is a call to a UE, the location information of the UE is first found in the network database, a paging message is then transmitted to all the cells in the location area (LA) where the UE last registered, and the call is finally connected by receiving a response from the UE. This location registration and paging procedure impose significant costs on radio channels. Therefore, performance modeling and cost analysis are essential aspects of identifying the optimal location registration method that can minimize the costs on radio channels [1,2].

To date, many location registration strategies have been proposed to reduce mobility management costs in mobile communication networks; representative methods include movement-based registration [2,3,4,5], distance-based registration [5,6,7,8], and zone-based registration (ZBR) [9,10,11,12,13]. In addition, even though there have been many studies on tracking area list-based registration for 4G/5G in recent years [14,15,16,17,18], this method is not a completely new one, but rather a flexible strategy that can be implemented in the same way as one of the traditional methods. Recently, distributed mobility management was introduced to overcome the inevitable obstacles of the existing 4G/5G, but it is still at a conceptual stage because a new design must be introduced [19]. Among the methods mentioned, the method adopted most often is zone-based registration because of its good performance and easy implementation [10,11,12,13].

This study has considered the zone-based registration method. The ZBR method is mostly implemented with one-zone-based registration (1ZR, ZBR with one zone) in which the number of zones (or LA) to be stored is one; however, there is also a possible implementation wherein the number of zones to be stored is two or more. In the case of 1ZR, location registration occurs frequently, as it is performed whenever a UE moves to a new zone. The two-zone-based registration (2ZR, ZBR with two zones) method has been proposed [9,10,11,12] to solve this method, and it is already known that 2ZR is superior to 1ZR in most cases.

However, existing studies comparing the performances of the two methods have a significant problem: Some studies on 2ZR [9,11,12] have proposed a mathematical model to analyze the performance of a UE with two zones and compared the performance with that in which the number of zones is one. The reported studies have also demonstrated that the performance with the case where the number of zones is two is typically better. However, as per the published data, when an incoming call arrives and paging is performed, it is assumed that the zone is paged simultaneously in 1ZR, whereas 2-step paging (2SP) is performed in 2ZR for the sake of performance comparison. As a result, a performance comparison was not performed properly under the same conditions. Therefore, even in the case of 1ZR (such as 2ZR), the performance should be compared accurately by considering 2-step paging.

Moreover, in previous studies, the performance of 2ZR was analyzed while assuming that it returned to the zone visited immediately before with probability Ө [9,11,12]. In this study, even in 1ZR, the paging cost under 2-step paging is calculated by reflecting the same mobility characteristics as in 2ZR, defining the states of the cells constituting the zone, and calculating the steady-state probability of each state. Ultimately, we aimed to determine exactly what the best method is under the same 2-step paging and by using the same mobility model.

This paper proceeds as follows: Section 2 introduces the zone-based registration and 2-step paging and suggests an inter-cell mobility model in 1ZR for comparison with 2ZR. In Section 3, the total signaling cost of 1ZR under 2-step paging is analyzed using the Markov chain model. Numerical results on the performance of 1ZR are obtained and compared with 2ZR in Section 4. Finally, Section 5 provides the conclusions of this paper.

2. Zone-Based Registration and 2-Step Paging

2.1. Zone-Based Registration

When the UE moves to a new zone, it registers the zone to the network database and stores it in the zone list in the UE. A case in which the UE only maintains one zone is referred to as 1ZR. As shown in Figure 1, when a UE moves from zone A to zone B, the UE stores zone B in the zone list and registers the location as zone B in the network database. Accordingly, the stored zone list of the UE and the network database happen to be the same. At this time, when a call is made to the UE, the UE connects the call by paging to all cells in the zone where the UE was last registered.

A UE may store one or more zones in a zone list. The case in which two zones are stored is called 2ZR. As shown in Figure 2, after the UE registers zone A, it enters register zone B, and when a call to the UE occurs while in zone B, the network pages zone B and connects the call.

By contrast, if the UE is in zone B and then returns to zone A, it does not register again, since the zone is already in the zone list. Therefore, the number of location registrations of 2ZR is less than that of 1ZR; however, the paging cost increases. If an incoming call occurs after the UE returns to zone A, the network first pages zone B—which was registered recently—since the network does not know that the UE is returning to zone A. If there is no response, the call is connected by paging zone A. As such, 2ZR uses a 2-step paging method. Nevertheless, the paging cost is bound to increase compared to that of 1ZR. In summary, when 2ZR is adopted, the location registration cost is reduced compared to 1ZR, but the paging cost is increased.

2.2. 2-Step Paging

When a call to the UE occurs, this incoming call is connected through the paging procedure. Most mobile communication networks adopt a simultaneous paging method in which a paging message is issued to all of the cells within a corresponding zone. Although this method has the advantage of immediately receiving the acknowledgment of the paging, there is a problem in terms of efficient use of the radio channels because the paging message is transmitted to all the cells in the zone simultaneously. To improve this strategy, 2-step paging [9,11,12], in which the zone is divided into two small areas and paging is performed sequentially over two times, can be considered. In this case, if the UE responds in the first paging on the first small area, it is not necessary to perform the second paging on the remaining small area, which reduces the paging cost compared to that of conventional simultaneous paging. In the 2ZR studies, the paging cost was calculated by assuming that the paging was performed by 2-step paging. By contrast, in 1ZR, the performance was compared with that of 2ZR by still applying the method of paging the entire zone simultaneously. As a result, performance comparisons were not performed properly under similar conditions. Therefore, even in the case of 1ZR, it is necessary to accurately compare performance by considering 2-step paging in a manner similar to 2ZR [9,11,12].

In this study, we assume that the zone of 1ZR is composed of n cells, and it is divided into two small areas to obtain the optimal paging cost when 2-step paging is performed.

2.3. Mobility Model for Movement among the Zones

Assume that the zone of zone-based registration is a rectangle. In this case, the UE moves to one of the four adjacent zones.

As shown in Figure 3, the probability of moving to one of the four zones adjacent to the current zone—top, bottom, left, or right—is assumed to be equal to ¼. This mobility model is called a symmetric random walk model and is widely used for convenient analysis.

By contrast, in studies dealing with 2ZR, performance analysis has been performed based on the assumption that the probability of returning to the zone visited immediately before (θ) is different from the probability of entering one of the remaining zones ((1 − θ)/3) [9,12,13], as shown in Figure 4. Even in real situations, the probability of returning to the zone visited immediately before would be greater than that of symmetric random walk (θ > 1/4); hence, this assumption was validated to some extent. Although mathematical analysis became very difficult due to the introduction of the probability of returning to the zone visited immediately before (θ), it was shown that the higher θ was proportional to a lower location registration cost, and 2ZR was ultimately identified to be better in most cases. However, θ does not necessarily have to be greater than 1/4, and analysis is also possible for θ less than 1/4. Most importantly, by introducing θ, analysis is possible no matter what value θ has, including the symmetric random walk model.

In this study, we reflected the same movement characteristics as in 2ZR on 1ZR to allow for a comparison with 2ZR under similar circumstances by setting up a proper inter-cell mobility model in a zone composed of n cells. Finally, we calculated the optimal paging cost of 1ZR under the 2-step paging method by obtaining the steady-state probability of each cell through the Markov chain model.

Similar to 2ZR, if 2-step paging is adopted in the case of 1ZR, the zone would be divided into two small areas to perform paging. Therefore, to determine the cells that should be selected from which to start paging, we need to calculate the exact probability of being in each cell. However, to calculate the probability of being in each cell, it is first necessary to model the movement characteristics between the cells in the zone. As can be seen in the inter-zone mobility model in the studies on 2ZR, unlike the symmetric random walk model, the probability of moving to the zone visited immediately before (θ > 1/4) is different from the probability of moving to a new zone ((1 − θ)/3). These characteristics must be reflected in the movement between the cells in the zone in 1ZR.

2.4. Mobility Model for Movement among the Cells in a Zone

In this study, the movement characteristics between the cells in the zone are assumed to be as follows: In 1ZR, we assume that the UE in any cell enters the cell in the direction of the zone visited immediately before with probability q, and that it equally enters the remaining cells with probability r = (1 − q)/3, which is very similar to the movement characteristics between zones in 2ZR [9,12,13]. Note that q and θ have essentially the same properties and that q is just a cell version of θ. For example, in Figure 5, assuming that a UE enters a new zone from the left zone, the UE in cell a returns to cell p with probability q, and it enters the remaining adjacent cells with equal probability r = (1 − q)/3. Similarly, the UE in cell b enters cell a with probability q, and it enters the remaining adjacent cells with equal probability r = (1 − q)/3.

In Figure 5, assuming that the UE enters a new zone out of the left zone, the state of two cells adjacent to the zone visited just before is defined as 1. These two cells are defined in a similar manner to state 1 because the probabilistic characteristics are the same regardless of whether the cell is above or below. A UE belonging to state 1 enters the cell of the zone visited immediately before with probability q and enters the remaining cells with probability r = (1 − q)/3 in the same way. Even when the UE stays in a cell and is not in contact with the zone visited immediately before (state 2) and enters a neighboring cell, it enters the cell (state 1 cell) in the direction of the zone visited immediately before with probability q, or it enters the remaining cells with probability r = (1 − q)/3 in the same way. Two cells that are not in contact with the zone visited immediately before are defined as the same state 2 because the probabilistic characteristics are the same regardless of whether the cell is above or below.

Likewise, even when the total number of cells is 9 (n = 9), it is assumed that all cells enter the cell in the direction of the zone visited immediately before with probability q and enter the remaining cells with probability r = (1 − q)/3. Among the three cells in contact with the zone visited immediately before, the upper and lower cells have the same probabilistic characteristics, so they can be defined as the same state 1. However, the middle cell has different probabilistic characteristics from the cells above and below it, so it becomes a new state 2; this is also true for state 4 cell and state 6 cell. As a result, as shown in Figure 6, the cells are classified into six states.

In the case of n = 4² = 16, n = 5² = 25, …, the state can be defined to analyze the performance similarly.

3. Analysis of ZBR under 2-Step Paging

In this section, let us analyze the performance of 1ZR when 2-step paging is adopted by using a Markov chain model. A Markov chain is defined as a process that the conditional distribution of any future state X_n+1, given that the past states X₀, X₁, ⋯, X_n−1, and the present state X_n, are independent of the past states and depend only on the present state X_n [20]. In this study, since the next cell depends only on the present cell, we can use a Markov chain model to analyze the performance. To evaluate the total signaling cost on radio channels, we define the following notations:

• U: location registration cost for one registration
• C_R: location registration cost per unit time
• V: paging cost for one cell
• C_P: paging cost per unit time
• T_c: interarrival time of calls (r.v., T_c ~ Exp[λ_c], E(T_c) = 1/λ_c)
• T_m: time spent in a cell (r.v., E(T_m) = 1/λ_m)
• $f_m^{*}\left(s\right)$: Laplace–Stieltjes transform for T_m (=${{\displaystyle \int }}_{t=0}^{\infty }{e}^{-st}{f}_m\left(t\right)dt$.
• R_m: remaining time from the call occurrence to the time the UE moves out of the cell (r.v.).

3.1. 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.

Each element p_ij of the state transition matrix 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 1ZR is π_i, then π_i can be calculated using the following balance equations [20].

$$\pi =\pi P, {{\displaystyle \sum }}_i{\pi }_i=1 \left(i=1, 2, 3,\dots \right)$$

(1)

For example, when n = 4, the steady-state probability can be calculated using the following equations.

$$\left[\begin{array}{cc}{\pi }_1& {\pi }_2]\end{array}=\right[\begin{array}{cc}{\pi }_1& {\pi }_2]\end{array} \left[\begin{array}{cc}q+2r& r\\ q+2r& r\end{array}\right], {\pi }_1+{\pi }_2=1$$

(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 (p₁₂) or state 3 (p₁₃) is r, and the transition probability from state 1 to state 1 (p₁₁) 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 = 4² = 16, n = 5² = 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 1ZR under 2-step paging, let us compare the performance with 2ZR. To apply 2-step paging in 1ZR, a zone composed of several cells should be considered; however, in the previous study, only the movement between the zones was considered. Therefore, to compare the performance with the previous 2ZR that only considered the time spent in the zone, it is also necessary to determine the time spent in the zone in the 1ZR.

3.2. Analysis of Time Spent in a Zone in 1ZR

To compare the performance with 2ZR, which only considers movement between the zones, let us also find the time spent in the zone for 1ZR.

First, let us determine the average number of cells until a UE goes through a zone in 1ZR with a zone composed of several cells and then find the time spent in the zone based on this number. Let us then find the average number of cells until a UE that has just entered the current zone passes through the zone. For example, consider the case of n = 9. The nine cells on the right are the ones in the current zone and the three cells on the left are the ones in the zone visited immediately before. If the case of leaving the current zone and performing location registration is defined as the absorbing state of state 0, then the number of cells that a UE that has just entered the current zone (states 1 or 2) enters until it leaves this zone can be obtained using the following method:

The state transition diagram and state transition probability matrix for the state 0 (absorbing state) that enters the new zone and performs location registration and six states of nine cells in the current zone are as follows:

In Figure 9, R represents the transition probability matrix from the transit states to the absorbing state, T represents the transition probability matrix between the transient states, and I represents the identity matrix. Starting from the current state i, let E_ij be the average number of visits to state j before entering the absorbing state. Then, E—a matrix of E_ij—can be obtained as follows ([20], p. 231):

E = (I − T)⁻¹

(3)

Defining N_i as the number of transitions (number of cells visited) from the transit state i to the absorbing state 0, when n = 9, the average number of cells passing through in a zone, E(N), can be expressed as follows:

E(N) = p₁ × E(N₁) + p₂ × E(N₂)

(4)

where p_i (i = 1, 2) is the probability that, among the UEs that have just entered the current zone, the state is i. E(N_i) can be obtained by adding all the values of row i of E = (I − T)⁻¹.

$$E(N_{\mathrm{i}})={{\displaystyle \sum }}_{j=1}^6{E}_{ij}^{} \mathrm{where} E={(I-T)}^{-1}$$

(5)

Suppose A_i (i = 1, 2) is the event that a UE enters the current zone from the zone visited immediately before and becomes the state i and that B_i (i = 1, 2, …, 6) is the event that a UE is in the cell corresponding to the state i (in the zone visited just before). Then, considering states in Figure 10, we can obtain the following equation:

$$\begin{gathered} P\left[A_1\right]=P\left[B_1\right] \times P[\mathrm{location} \mathrm{registration} \mathrm{occurs} \mathrm{with} \mathrm{one} \mathrm{cell} \mathrm{entrance}|B_1] \\ +P[B_5] \times P[\mathrm{location} \mathrm{registration} \mathrm{occurs} \mathrm{with} \mathrm{one} \mathrm{cell} \mathrm{entrance}\left|B_5\right] \\ ={\pi }_1\times (q+r)+{\pi }_5 \times 2r \end{gathered}$$

(6)

$$\begin{gathered} P\left[A_2\right]=P\left[B_2\right] \times P[\mathrm{location} \mathrm{registration} \mathrm{occurs} \mathrm{with} \mathrm{one} \mathrm{cell} \mathrm{entrance}|B_2] \\ +P[B_3] \times P[\mathrm{location} \mathrm{registration} \mathrm{occurs} \mathrm{with} \mathrm{one} \mathrm{cell} \mathrm{entrance}\left|B_3\right] \\ +P\left[B_6\right] \times P[\mathrm{location} \mathrm{registration} \mathrm{occurs} \mathrm{with} \mathrm{one} \mathrm{cell} \mathrm{entrance}|B_6] \\ ={\pi }_2 \times q+{\pi }_3 \times r+{\pi }_6 \times r \end{gathered}$$

(7)

The average number of cells visited in the zone should be obtained by only considering when the location registration is performed for the first time in the current cell; therefore, each probability of becoming state 1 ($p_1)$ and state 2 ($p_2)$ is as follows:

$$p_1=\frac{P\left[A_1\right]}{P\left[A_1\left]+P\right[A_2\right]}=\frac{{\pi }_1\times \left(q+r\right)+{\pi }_5\times 2r}{{\pi }_1\times \left(q+r\right)+{\pi }_5\times 2r+{\pi }_2\times q+{\pi }_3\times r+{\pi }_6\times r} .$$

(8)

$$p_2=\frac{P\left[A_2\right]}{P\left[A_1\left]+P\right[A_2\right]}=\frac{{\pi }_2\times q+{\pi }_3\times r+{\pi }_6\times r}{{\pi }_1\times \left(q+r\right)+{\pi }_5\times 2r+{\pi }_2\times q+{\pi }_3\times r+{\pi }_6\times r}.$$

(9)

Therefore, the average number of cells visited in the zone from state 1 and state 2 until moving to another zone can be calculated using the follow equation:

$$E(N)=E\left(N_1\right)\times p_1 + E\left(N_2\right)\times p_2$$

(10)

where E(N₁) represents the average number of cells visited from state 1 until absorbing to state 0 and E(N₂) represents the average number of cells visited from state 2 until absorbing to state 0.

Finally, when the time spent in a cell follows an exponential distribution with a mean 1/${\lambda }_m$ [9,16], the average time spent in a zone, E[T_Z], can be obtained as follows:

$$E[T_Z]=E\left(N\right)\times \frac{1}{{\lambda }_m}$$

(11)

3.3. Total Signaling Cost

The total signaling cost on radio channels is composed of the registration cost and the paging cost. In 1ZR, the location registration cost per unit time can be calculated as follows:

$$C_R^{1ZR}=U\times {\lambda }_m\times P_R=U\times {\lambda }_m\times \left[{{\displaystyle \sum }}_{i=1}^K({\pi }_i\times p_{i0})\right]$$

(12)

• $U$: location registration cost for one registration
• P_R: the probability that a UE performs location registration when it leaves the current cell
• K: total number of states in a zone

Note that, in 2ZR, the location registration cost per unit time can be calculated as follows:

$$C_R^{2ZR}=U\times \left(1-\theta \right)\times \frac{1}{E\left[T_Z\right]}$$

(13)

• $U$: location registration cost for one registration
• $E[T_Z]$: the expected value of time spent in a zone

In this study, a 2-step paging method has been used to perform paging by dividing a zone stored in the network to which the UE belongs into two sub-areas. When the incoming call to a UE arrives, the paging cost can be obtained by multiplying the paging cost per cell by the number of cells to be paged.

The cells located at the boundary at which there is a probability of returning to the zone visited immediately before are called first sub-area A1, and the other cells are called the second sub-area A2.

When a call to a UE arrives, the first sub-area A1 adjacent to the zone visited immediately before should be paged, and the next sub-area A2 should be paged if there is no response. Therefore, the paging cost per unit of time can be obtained by multiplying the paging cost per cell by the number of cells to be paged as follows:

$$C_P^{1ZR}=V{\lambda }_c\left[n_1{{\displaystyle \sum }}_{i=1}^{k^{*}}{\pi }_i+\left(\left(n_1+n_2\right){{\displaystyle \sum }}_{i=k^{*}+1}^K{\pi }_i\right)\right]$$

(14)

• $V$: paging cost per cell
• ${\lambda }_c$: number of incoming calls per unit time
• $n_i$: number of cells in i-th sub-area $A_i$
• K: total number of states in a zone
• k*: total number of states for the cells of the first sub-area to be paged
• π_i: the probability that a UE is in state i

In the above, k*, the total number of states for the cells of the first sub-area to be paged, is k that makes the paging cost, $V{\lambda }_c\left[n_1{\displaystyle \sum }_{i=1}^{k^{*}}{\pi }_i+\left(\left(n_1+n_2\right){\displaystyle \sum }_{i=k^{*}+1}^K{\pi }_i\right)\right]$, minimal for every k, k = 1, 2, …, K.

Since the total signaling cost on radio channels consists of location registration cost and paging cost, the total signaling cost per unit time of 1ZR can be calculated as the sum of location registration cost and paging cost, as follows:

$$C_T^{1ZR}=C_R^{1ZR}+C_P^{1ZR}$$

(15)

4. Numerical Results

To compare the performances of 1ZR and 2ZR in various environments using the analytical model presented in Section 3, the following factors are assumed [9,10,11,12]:

▪

The zone is composed of cells of the same square.

▪

It is assumed that the probability of moving to the zone visited immediately before, q, is 0.3.

▪

The time spent in a cell follows an exponential distribution with a mean of 1/λ_m.

▪

The number of incoming calls to a UE follows a Poisson distribution with rate λ_c.

▪

The location registration cost is four times the paging cost per cell (U = 4, V = 1).

For 1ZR and 2ZR, the total signaling cost for various numbers of cells in a zone when simultaneous paging (1SP) or 2-step paging (2SP) are performed is shown in Figure 11. When the number of cells in the zone is n = 4, all the cases have the minimal cost, but among them, 1ZR under 2-step paging has the lowest signaling cost. Specifically, it can be seen that 1ZR under simultaneous paging exhibits worse performance than 2ZR, but 1ZR under 2-step paging has better performance than 2ZR. Note that it can be seen that the simulation results for proposed 1ZR with 2SP are generally consistent with the numerical results using proposed mathematical models and equations.

Figure 12 shows the paging cost, location registration cost, and total signaling cost of 1ZR and 2ZR. By examining the total signaling cost of both methods, we can see that the cost is the smallest when n is 4 and the total signaling cost of 1ZR is less than that of 2ZR. The main reason for this is that, in the case of 1ZR, selecting 2SP increases the probability of success in the first paging, so the number of cells to be paged is significantly reduced compared to 2ZR.

When the number of cells in the zone is n = 1, the total signaling cost of 1ZR is higher than that of 2ZR due to the very high registration cost. However, a decrease in the rate of registration cost in 1ZR and an increase in the rate of paging cost in 2ZR demonstrate steepness as the number of cells in the zone increases. Consequently, the total signaling cost of 1ZR is less than that of 2ZR in most of the cases when n is large.

Figure 13 shows the total signaling cost for two values of q under different numbers of entering cells in unit time (λ_m). The total signaling costs of 1ZR and 2ZR increase as the number of entering cells increases because the probability of location registration increases. Generally, when q = 0.5 in the case of 2ZR, the total signaling cost is lower than that of q = 0.3, because as q increases, the probability of location registration decreases. By contrast, in the case of 1ZR, when q = 0.5, the total signaling cost is large compared to q = 0.3, because location registration occurs frequently due to the high probability of moving to the previous cell. As a result, when q = 0.3, the total signaling cost is always low compared to q = 0.5. 1ZR shows a lower total signaling cost than 2ZR when the number of entering cells is small. However, as the number of entering cells increases, the total signaling cost of 1ZR increases more steeply and exceeds that of 2ZR due to frequent location registration.

In conclusion, 1ZR with small q shows the best performance when the number of entering cells is small, and 2ZR with large q shows the best performance when the number of entering cells is large.

Figure 14 shows the total signaling cost of 1ZR for two values of q under simultaneous paging or 2-step paging. From the figure, it can be seen that 2-step paging is superior to simultaneous paging and that the difference is very large. We can also see that the larger q is associated with the larger total signaling cost. In this case, if the same paging method is adopted, then the total cost of having a large q is high because location registration occurs more frequently due to an increase in the probability of returning to the previous cell. In conclusion, it can be noted that if 2-step paging is adopted, then when n = 4, the total signal cost is minimized and the paging cost is significantly reduced compared to simultaneous paging. Moreover, the total signal cost is significantly reduced.

5. Conclusions

In this study, we have considered zone-based registration under 2-step paging. 2ZR (ZBR with two zones) is known to outperform 1ZR (ZBR with one zone) in most cases. However, in studies on comparison of 1ZR and 2ZR, the paging method of 1ZR was simultaneous paging and that of 2ZR was 2-step paging.

Moreover, we have analyzed the performance of 1ZR when 2-step paging (2SP) is applied in the same manner as 2ZR for exact performance comparison under similar conditions. To analyze the performance of 1ZR under 2-step paging, we have considered the zone that is composed of n cells and presented the mobility model for movement between the cells in a zone. The steady-state probability of each cell was obtained by using the Markov chain model, and the total signaling cost was finally calculated. Through various numerical results, it could be seen that when 1ZR adopts 2-step paging such as 2ZR, 1ZR behaves in a manner superior to 2ZR in most cases, which is attributed to the reduction in the paging cost.

In conclusion, when 2-step paging can be applied to both 1ZR and 2ZR, it is possible to reduce the paging cost and finally reduce the total signaling cost by selecting and operating the better 1ZR and 2ZR while considering the change in the traffic environment. In this study, we adopted a rather simple inter-cell mobility model and assumed that the time spent in a cell follows an exponential distribution for its simplicity. In the future, studies on diverse inter-cell mobility models and various types of distributions on time spent in a cell or a zone will be performed for a more sophisticated comparison between 1ZR and 2ZR.