## 1 Introduction  
To successfully connect incoming calls to the correct user equipment (UE), wireless communication networks use methods that track the current location of the UE. Many wireless resources are utilized for these mobility management methods [1; 2; 3; 4; 5]. Mobility management consists of two operations: location registration and paging [1; 6; 7]. Location registration is a process wherein the network database modifies the current location of the UE each time the UE moves to a new registration area (RA) to report its current location to the network. In that system, the RA is composed of several zones, where each zone is comprised of several cells that do not overlap with each other [8; 9]. Paging is the process of sending a paging message to all zones that are included in the RA to which the UE belongs whenever an incoming call arrives. Therefore, depending on the size of the RA (that is, the number of zones inside an RA), the numbers of registrations and paging messages vary according to the UE's movement, leading to substantial changes in the cost of wireless signaling traffic [4; 10; 11; 12; 13].  
To date, many schemes have been proposed to minimize signaling costs at radio frequencies, which can be classified into zone-based [1; 8; 14; 15; 16; 17], distance-based [18; 19],movement-based [20; 21], timer-based [1; 4; 22; 23], and so on [9; 13; 24; 25]. Among these schemes, a zone-based registration scheme is employed by most wireless communication networks due to its good performance and convenient implementation [1; 5]. Nevertheless, the biggest disadvantage of a zone-based method is the registration traffic generated by UEs that frequently enter and exit the RA boundary. To improve the performance when only one zone is included in RA (1Z), systems having two zones (2Z) or three zones (3Z) have been proposed [15; 16; 17]. Those studies evaluated the performances of 2Z and 3Z while assuming a square-shaped zone. However, considering the actual situation, it is difficult to say that a square best represents the shape of the zone. Rather, many studies have used more hexagonal zones that extend widely adopted hexagonal cells [3; 8; 10; 18; 21]. In this study, we analyzed the performances of 1Z, 2Z, and 3Z for a hexagonal-shaped zone by creating exact models of two types of 3Z structures in more realistic wireless network environments.  
This study focused on the zone-based registration method and evaluated the performance of the 3Z system in the form of a hexagonal-shaped zone. To analyze the performance of the 3Z, the semi-Markov process theory [26] was used to obtain the steady-state probability and compare its performance with those of 1Z and 2Z systems.  
The rest of this paper proceeds as follows: Section 2 defines all feasible states when three hexagonal zones are operated as one RA. This section also mathematically models the transition probability between states, the probability that registration will be required during a state transition, and the number of paging messages. In Section 3, the steady-state probability is obtained from the state transition probability, after which the registration and paging cost incurred per UE are computed using steady-state probability. Subsequently, Section 4 compares the performances of 1Z, 2Z, and 3Z systems under realistic parameter values. Finally, the conclusion is presented in Section 5.