## 4 Numerical Analysis
### Intra-Domain Handoff for Vehicles in PMIPv6 with Authentication
The binding update cost (\(BUC_{\text{PMIPv6}}^{\text{VANET}}\)) concerns the cost of constructing a special, two-way tunnel connecting the MAG and LMA, \(T_{\text{Trumel-Establishment}}\), as stated in Equation (6). This tunnel is then used to exchange the data packets destined for the vehicle from a correspondent node (CN) or vice versa. If the given vehicle's entry cannot be in the BCE, it then performs the registration of a new MAG; otherwise, a BCE lookup for the vehicle is performed. Each connected vehicle has an entry stored in the BCE that contains information on the vehicle's ID, its home network prefix (HNP), a proxy care-of-address(p-CoA), and the IP address of the associated MAG. The \(BUC_{PMIP6}^{VANET}\) in VANET is expressed as follows in Equation (6). \[BUC_{PMIP6}^{VANET}=T_{Tummel-Establishment} \tag{6}\]  
Building a bidirectional tunnel between the MAG and LMA requires the transmission of two control messages. One of these messages is a PBU from the MAG to LMA, and the other is a PBA from the LMA to MAG. As a result, it costs twice as much for network entities to exchange a control packet \((T_{MAG-LMA})\). The control packet's size, \(S_{Control}\), depends on the mobility option, mobility header, and IPv6 header size. Therefore, the tunnel establishment cost, \(T_{Tummel-Establishment}\), is expressed as follows in Equation (7). \[T_{Tummel-Establishment}=S_{Control}\times 2T_{MAG-LMA} \tag{7}\]  
After the CN and vehicle have established a secure connection, packet exchange can begin. A packet produced by the CN is transferred to the LMA in the first phase. The LMA looks for a BCE entry for the intended vehicle and any associated MAGs. When it locates them, the LMA sends the packet across a secure bidirectional tunnel to the related MAG, and, finally, the MAG transmits the packet to the designated vehicle. Data packets, \(S_{Data}\), are sent across wired and wireless networks. For ease of use, wireless connections are taken for the CN to the MAG and the MAG to the vehicle. Wired links between the MAG and LMA are taken into consideration in the same reference. Given that wireless networks are unstable and have a major impact on the cost of packet delivery, here, \(\mu\) is used to represent the unit cost of packet transmission over a wired link, and \(\tau\) is taken to represent the unit cost of packet transmission over a wireless link. A packet's transmitting cost could be specified as in Equation (8). \[PDC_{PMIP6}^{VANET}=S_{Data}\times\left(\mu\times T_{CN-MAG}+\tau\times 2T_{ LMA-MAG}+\mu\times T_{MAG-Vth}\right)+T_{P} \tag{8}\]  
where the processing cost, Tp, as expressed in Equation (9), comprises the lookup cost in the BCE; in Equation (9), \(\beta\) represents the unit cost of lookup at the LMA. \[T_{P}=\beta\times\log(N_{MAG}\times N_{Vth/MAG}) \tag{9}\]  
Thus, Equations (8) and (9), when combined, give the cost of sending a packet \(PDC_{PMIP6}^{VANET}\) as expressed in Equation (10). \[T_{\text{Authentication}}^{\text{VANET}}=T_{\text{Network-Entities}}^{\text{ VANET}}+T_{\text{Mobile-Entities}}^{\text{VANET}} \tag{2}\]0 \[T_{\text{Authentication}}^{\text{VANET}}=T_{\text{Network-Entities}}^{\text{ VANET}}+T_{\text{Mobile-Entities}}^{\text{VANET}} \tag{2}\]1  
Therefore, Equations (5), (7) and (10), give rise to Equation (11) as the total cost (\(TC_{PMIP6}^{VANET}\)) of packet delivery in PMIP6 for VANET [23].