## 3 Problem Statement
The problem occurs in a scenario when the forwarder (the node with higher priority) does the broadcasting of the packet in its own transmission zone after the expiry of the timer value, while some PFNs of the source node are not in the range (located in reproducingregions, as shown in Figure 2), and hence do not receive a copy of the original packet. This results in the generation of duplicate packets. The source node S broadcasts the packet, which is received by the nodes \(P_{fn}^{1}\), \(P_{fn}^{2}\), \(P_{fn}^{3}\), \(S_{p}^{1}\), \(S_{p}^{2}\), \(S_{p}^{3}\),\(P_{g}^{1}\), \(P_{g}^{2}\), and \(P_{g}^{3}\), as shown in Figure 2. The suppressed nodes due to being deeper than the source node, directly drop the packet but \(P_{fn}^{1}\), \(P_{fn}^{2}\), \(P_{fn}^{3}\), \(P_{g}^{1}\), \(P_{g}^{2}\), and \(P_{g}^{3}\) calculate the holding time value and set the timers. Based on the parameters of the holding time, the highest priority node should be node \(P_{fn}^{1}\). The nodes \(P_{g}^{1}\), \(P_{g}^{2}\) and \(P_{g}^{3}\) are out of the transmission range of the highest priority node (\(P_{fn}^{1}\)). Therefore, they will generate duplicate packets and transmit these packets, resulting in higher energy consumption. On the contrary, EESEVBF uses control packets to suppress the reproducing nodes, as discussed below. The propagation delay \(\tau(n1,n2)\) between two nodes in UWSNs is proportional to the separation distance between them, as shown in Figure 3. Node A can suppress the broadcasting of node B when the holding time difference between them is greater than the propagation delay and in that case, the suppressed otherwise duplicate region will be achieved. The reproducing region is achieved due to the hidden terminal problem. In ESEVBF, duplicate packets are initiated and forwarded in both suppressed and reproducing regions, while in EESEVBF, this is achieved by using the control packets. Mathematically, the suppressed region, duplicate region and reproducing region can be expressed in Equations (1)-(3) respectively:  
\[HT_{p}^{n1}-HT_{p}^{n2}>\tau(n1,n2)\wedge Tr_{n1}\geq D_{n1}^{n2}<Tr_{S} \tag{1}\]  
\[HT_{p}^{n1}-HT_{p}^{n2}<\tau(n1,n2)\wedge Tr_{n1}\geq D_{n1}^{n2}<Tr_{S} \tag{2}\]  
\[HT_{p}^{n1}-HT_{p}^{n2}>\tau(n1,n2)\wedge Tr_{n1}\leq D_{n1}^{n2}<Tr_{S} \tag{3}\]  
The duplicate region can be easily avoided if the holding time difference is larger, but at the same time, it can generate a long end-to-end delay. ESEVBF estimates the holding time in such a way that the suppression is achieved frequently with the tradeoff of end-to-end delay. Figure 1: Forwarder Node Selection Scenario. The proposed EESEVBF scheme uses the following notations:  
_Sink Node S\({}_{k}\):_ Sink node in UWSN is a superficial node that is placed at the sea surface to collect data from the deployed nodes inside the water. A radio link is used by the sink nodes for communication. The packet received by the sink node is considered to be a successful conveyance to its terminus point. These superficial nodes are either steady at their starting points or they dynamically move in their planes. Let \(S_{k}\) be the set of sinks in the network, then  
\[S_{k}=S_{k1},S_{k2},S_{k3},\ldots S_{kn} \tag{4}\]  
where \(S_{ki}\) is a specific node and n is the total number of sinks. _Transmission Range (\(T_{r}^{i}\)) of Node(i)_: The transmission range of any node i(\(x_{s},y_{s},z_{s}\)) is the omnidirectional distance up to where it can transmit the data packet. _Eligible Neighbors of Node(i) (\(EGN_{i}\))_: The nodes that are in the transmission range of Node (i) are the eligible neighbors of Node (i). This can be mathematically expressed as:  
\[EGN_{i}=i\epsilon S_{N}\wedge Dist_{i}^{j}\leq T_{r}^{j1} \tag{5}\]  
where \(S_{N}\) is the total numbers of nodes in the network and \(Dist_{i}^{i}\) is the Euclidean distance between node \(i(x_{i},y_{i},z_{i})\) and \(j(x_{i},y_{i},z_{i})\). Figure 2.0  
Figure 2.1 Hidden Terminal Problem Scenario.