## 4 Numerical Analysis
### Intra-Domain Handoff for Vehicles in PMIPv6 with Authentication  
The total cost of packet delivery (\(T_{\text{C}_{PMIPv6}^{\text{VANET}}}\)) in VANET in particular has three components, namely the authentication cost (\(T_{\text{Authentication}}^{\text{VANET}}\)) with a trusted third party (TTP), the binding update cost (\(BUC_{\text{PMIPv6}}^{\text{VANET}}\)), and the packet delivery cost (\(PDC_{\text{PMIPv6}}^{\text{VANET}}\)). Therefore, Equation (1) represents the total packet delivery cost (\(TC_{\text{PMIPv6}}^{\text{VANET}}\)) as the sum of these three component costs. \[TC_{\text{PMIPv6}}^{\text{VANET}}=T_{\text{Authentication}}^{\text{VANET}}+BUC _{\text{PMIPv6}}^{\text{VANET}}+PDC_{\text{PMIPv6}}^{\text{VANET}} \tag{1}\]  
The authentication of each vehicle along with the network entities is crucial in ensuring secure communication. As a result, the authentication procedure includes both the authentication of vehicles and network entities such as the MAG and LMA. Additionally, it was presumed that each PMIPv6 domain would have a trusted third-party (TTP) server in place, making the TTP responsible for authenticating each entity. Furthermore, it is believed that all TTPs are linked and situated one hop apart. Thus, the authentication cost, \(T_{\text{Authentication}}^{\text{VANET}}\), is expressed as in Equation (2) as the sum of the cost of the authentication of network entities and the cost of the authentication of mobile entities such as vehicles. \[T_{\text{Authentication}}^{\text{VANET}}=T_{\text{Network-Entities}}^{\text{ VANET}}+T_{\text{Mobile-Entities}}^{\text{VANET}} \tag{2}\]  
Network entities and the TTP server will exchange authentication control messages, \(S_{\text{TTP}}\), throughout the course of the authentication procedure. The PMIPv6 domain contains several MAGs; each MAG will send a control message to the TTP server for authentication, and the TTP server will respond by sending the control message back to the MAG. In the same way, the LMA sends the control message to the TTP, and, in response, the TTP will send a control message back to the LMA. Hence, the cost of the authentication of network entities \(T_{\text{Network-Entities}}^{\text{VANET}}\) can be as stated in Equation (3). \[T_{\text{Network-Entities}}^{\text{VANET}}=N_{MAG}\times(S_{\text{TTP}}\times 2 T_{\text{MAG-TTP}})+S_{\text{TTP}}\times 2T_{\text{LMA-TTP}} \tag{3}\]  
Secure communication requires the validation of every vehicle. A link between the vehicle and the MAG must be created to authenticate a vehicle, and this requires the setup cost, \(T_{\text{Setup}}\). The control message, \(S_{\text{TTP}}\), is sent by the MAG with the authentication request to the TTP server. As a result, each vehicle in the PMIPv6 domain will send this control message to the TTP server through the MAG, and once the authentication is complete, the TTP server will send the authentication control message back to the MAG. Hence, the authentication cost of mobile entities (\(T_{\text{Mobile-Entities}}^{\text{VANET}}\)) is as represented in Equation (4). \[T_{\text{Mobile-Entities}}^{\text{VANET}}=N_{\text{Velb}}\times\big{(}T_{ \text{Setup}}+Max(S_{\text{TTP}}\times 2T_{\text{MAG-TTP}},S_{\text{TTP}} \times T_{\text{TTP-LMA}})\big{)} \tag{4}\]  
Therefore, from Equations (3) and (4), the authentication cost \(T_{\text{Authentication}}^{\text{VANET}}\) expressed in Equation (2) can be characterized as in Equation (5). \[T_{\text{Authentication}}^{\text{VANET}}= N_{MAG}\times(S_{\text{TTP}}\times 2T_{\text{MAG-TTP}})+S_{\text{ TTP}}\times 2T_{\text{LMA-TTP}}+N_{\text{Velb}} \tag{5}\] \[\times\big{(}T_{\text{Setup}}+Max(S_{\text{TTP}}\times 2T_{\text{ MAG-TTP}},S_{\text{TTP}}\times T_{\text{TTP-LMA}})\big{)}\]  
Once authentication is complete, the TTP server multicasts the authentication response to the corresponding LMA, as well as the TTPs of neighboring PMIPv6 domains, for inter-domain handover.