## 3 Proposed Scheme
### Estimation of \(HT_{p}^{i}\)  
Every node \(i\) that is within the \(PFZ\) first computes its holding time \(HT_{p}^{i}\), when it receives the packet \(p\) from \(S\) as follows:  
\[HT_{p}^{i}=\alpha+\beta+1-\left(\frac{D_{D}^{S}-D_{D}^{i}\cos(\theta_{i})}{T_{ r}^{S}}\right) \tag{4}\]  
The first factor of \(HT\) expression, \(\alpha\), considers the distance of potential forwarder from the edge of the \(T_{r}^{S}\) that is scaled with the inverse normalized residual energy of the node. Any node that is closest to the edge of the \(T_{r}^{S}\) and has the maximum residual energy will be logically preferable forwarder and \(\alpha\) is computed as:  
\[\alpha=e^{(-E_{i})}\left(\frac{T_{r}^{S}-D_{S}^{i}}{v_{s}}\right) \tag{5}\]  
where  
\[E_{i} =\frac{e_{i}-e_{min}}{e_{max}-e_{min}}\] \[e_{min} =min\left(e_{j}|\forall j\in\chi_{i}\right)\] \[e_{max} =max\left(e_{j}|\forall j\in\chi_{i}\right)\] \[E_{i} \in[0,1]\]  
The energy of a node is relatively normalized to all the neighboring nodes' residual energy in that are neighbors of \(i\) and in PFZ. The node with maximum residual energy within the neighborhood, including the current forwarding node, will have the \(E_{i}=1\) and vice versa. In AHH-VBF, this factor increases the chances of node \(i\) to become potential forwarder if it is at the edge of the \(T_{r}^{S}\). On the contrary, the proposed scheme scales this parameter using the scaled residual energy of node \(i\). The \(e^{(-E_{i})}\) element in \(\alpha\) decreases overall \(HT_{p}^{i}\) of node \(i\) with larger residual energy and makes it more suitable candidate to forward \(p\). The next factor of the \(HT\), \(\beta\), is the ratio of the projection distance \(P_{i}\) of the potential forwarding node \(i\) from the centerline of the virtual cylinder with radius \(R\). This centerline connects nodes \(S\) and \(D\) that are at the center of the lower and upper faces of the cylinder. Nodes that are furthest from this centerline are not desirable as forwarders and their \(HT\) must be larger than the one that are closer the centerline, as shown in Figure 3. To achieve this, AHH-VBF just takes the ratio of \(P_{i}\) and \(R\), \(\beta=\frac{P_{i}}{R}\), which returns the value of \(\beta\) within the closed interval \([0,1]\). However, the value of \(\beta\) should be expanded to widen this value to easily avoid multiple data transmissions as:  
\[\beta=\tan\left(\frac{P_{i}}{R}\right) \tag{6}\]  
where \(P_{i}\) is estimated as:  
\[P_{i} =\left(2\times A\right)/D_{D}^{S},\] \[A =\sqrt{\rho\times\left(\rho-D_{D}^{S}\right)\times\left(\rho-D_{ D}^{i}\right)\times\left(\rho-D_{S}^{i}\right)},\] \[\alpha=e^{(-E_{i})}\left(\frac{T_{r}^{S}-D_{S}^{i}}{v_{s}}\right) \tag{5}\]0  
The last factor of the \(HT\), \(\gamma\), projects the distance of the potential forwarder towards the sink. Any node in \(PFZ\) that is closer to the sink is a suitable to be the next potential forwarder. The \(\gamma\) of all the nodes in \(PFZ\) is between \([0,1]\). In this factor, the element \(D_{D}^{i}\cos(\theta_{i})\) results the distance between the projection point \(q_{i}\) on the centerline and the sink node. Here, \(\theta_{i}\) is calculated as:  
\[\alpha=e^{(-E_{i})}\left(\frac{T_{r}^{S}-D_{S}^{i}}{v_{s}}\right) \tag{5}\]1  
The ratio of the difference between \(D_{D}^{S}\) and \(D_{D}^{i}\cos(\theta_{i})\) and \(T_{r}^{S}\) will be high when node \(i\) is closer to the sink and vice versa. Subtraction of that ratio from 1 will have a very small increment in the holding time of node \(i\) if it is closer to the sink node, enables node \(i\) to be more suitable forwarding candidate. On the hand, the holding time of node \(i\) will be sufficiently increased when it is far from the sink and closer to \(S\). In order to efficiently forward the data packet, multiple packets are exchanged between nodes to maintain 1-hop neighboring state at each node. These packets include neighbor request (**NEIGH_REQ**), neighbor acknowledgment (**NEIGH_ACK**), and **data** packet.