## 4 System Model
### Link Quality Estimation of Mobile Nodes  
The link quality of the channel is one of the key factors in the selection of the reliable route, which is estimated based on the Expected Number of Transmission (ETX) and Expected Any-path Count (EAX) parameters [59]. The ETX is the expected number of transmissions required before a packet is successfully delivered to the next-hop which also determines link quality and packet loss on both directions of a link. Node \(i\) is the intermediate node whose probabilities of receiving andforwarding data packets are \(\lambda_{i}\) and \(\mu_{i}\), respectively. These values are measured through the probe message that is transmitted using dedicated link before actual data transmission. Each node broadcasts a probe message to all of its neighbor nodes and maintains records of transmitted probe message for a designated period of \(\tau\) seconds. Accordingly, the packet delivery ratio probability from a source node to neighbor nodes is calculated as follows:  
\[\lambda_{i}=\frac{n_{w}}{w/\tau} \tag{9}\]  
whereas \(n_{w}\) denotes the number of probe messages delivered in a period of \(w\) seconds satisfying the condition \(w>\tau\). The probability that a neighbor's node \(i\) receives a packet from at least one forward node is \(p_{for}=1-\prod_{>s}\left(1-\mu_{i}\lambda_{i}\right)\), and the probability that node \(i\) successfully delivers a packet to at least one backward node is \(p_{back}=1-\prod_{d>i}\left(1-\mu_{i}\lambda_{d}\right)\). Consequently, the expected number of transmissions that node \(i\) needs to take can be calculated as:  
\[ETX=\frac{1}{p_{for}\times p_{back}}=\frac{1}{(1-\prod_{s>i}\left(1-\mu_{s} \lambda_{i}\right))(1-\prod_{d>i}\left(1-\mu_{i}\lambda_{d}\right))} \tag{10}\]  
The ETX calculates the expected number of transmissions needed to successfully deliver a packet from source to destination through the forward node set \(i\). The ETX metric selects all forwarded candidates and prioritizes them based on the Equation (10). However, the ETX metric also includes forwarded sets that increase packet overhead in the network. To address this issue, the EAX metric is employed to select only the optimal forwarded sets and prioritize them based on opportunistic routing. This decreases interference with neighbor's node and minimized packets overhead, and number of transmissions to some extent. The hop-count parameter is useful for optimized routing performance with minimum hop-count value. In addition to that, EAX minimizes the number of intermediate nodes and number of transmissions required for reliable packet's delivery on optimal route without contrarily affecting the performance of network. \(\text{C}^{s,\ d}\) defines the forwarded set of candidates from node \(s\) to \(d\) and \(\text{C}^{s,\ d}_{i}\) defines the node \(i\) forwarded set prioritizing range from 0 to 1. Therefore, the delivery probability between source and \(\text{C}^{s,\ d}_{i}\) is \(p_{for}\) and vice versa (considering both the forward data and backward ACK transmissions). Then, the EAX value from source to destination through forwarded set can be calculated as follows:  
\[EAX(s,\ d)=\frac{1+\sum_{i}EAX(\text{C}^{s,\ d}_{i},d)(1-\prod_{s>i}\left(1- \mu_{s}\lambda_{i}\right))\prod_{j-1}^{i-1}\!\!\left(1-\prod_{d>i}\left(1-\mu_ {s}\lambda_{j}\right)\right)}{(1-\prod_{s>i}\left(1-\mu_{s}\lambda_{i}\right))} \tag{11}\]  
In the above expression, the EAX metric selects and enables the potential candidate pairs of intermediate nodes based on the packet's delivery probability. The selection of potential candidates in the optimal route from source to destination nodes is performed as follows. Firstly, the ETX metric potential candidate \(\text{C}^{s,\ d}_{potential}\) is determined based on the best path, such that if \(ETX(s,d)>ETX(j,d)\), node \(j\) will be added to the potential candidates in the route. Secondly, the subset of potential nodes \(\text{C}^{s,\ d}_{potential}\) is selected as the actual candidates set \(\text{C}^{s,\ d}\) having the smallest ETX value to the destination.