## 4 System Model
### Node Mobility Estimation  
Node mobility estimation governs the unpredictable movement of a node in a network, which helps to estimate the probability of a link failure and link instability in advance. The random waypoint model (RWP) is extensively used for estimation of node's mobility and for simulation analysis in the mobile wireless network due to its simplicity and better QoS [58]. Several parameters, such as speed, move time, pause time, and separation distance of the mobile nodes, are configured for the estimation of node's mobility. The RWP model periodically estimates the mobility of nodes and selects the random destination point (\"waypoint\") in the network based on the above parameters. Once the nodes reach their selected waypoints, nodes are pause for defined duration of time (pause time), and this process is repeated. The waypoints are uniformly and randomly chosen in the network area. In the proposed scheme, nodes can move randomly in an area of operation and pause for an arbitrary period of time. The maximum and minimum velocity (\(v_{\max}\) and \(v_{\min}\)) of each node is configured based on the RWP model along with its calculation of pause time duration, so as to simulate real deployment scenarios. Network nodes are uniformly distributed in the RWP model between the velocities range \([v_{\max}-v_{\min}]\), where \(v_{\max}\) and \(v_{\min}\) are the minimum and maximum velocities of the nodes, respectively. The node speed distribution \(f_{(s,d)}(v)\) in the dynamic network is described as below:  
\[f_{(s,d)}(v)=p_{move}.\frac{1}{v\text{In}(v_{\max}/v_{\min})}+p_{pause}. \delta(v) \tag{6}\]  
where the velocity function \(v\) is defined in the range of \(v\in[v_{\max}-v_{\min}]\), \(p_{pause}\) and \(p_{move}=(1-p_{pause})\) are the probabilities of nodes in the pause and moving states in the network respectively. \(\delta(v)\) defines the Dirac delta function whose value updates according to the velocity of the node. The value of \(\delta(v)\) varies from 0 to 1, when the velocity of a node is maximum, \(\delta(v)\) is updated to 0, and when the velocity of the node is minimum, \(\delta(v)\) become 1. The pause time of nodes is considered to be \(t_{p}\geq 0\), and the value of probability of pause time \(p_{pause}\) can be calculated as follows:  
\[p_{pause}=\frac{t_{p}}{t_{p}+E[D]\frac{\text{In}(v_{\max}/v_{\min})}{(v_{\max }-v_{\min})}} \tag{7}\]  
where \(E[D]\) denotes expected value of distance cover due to node movement on the whole trip. If a node starts from the pause state \(p_{pause}\), it pause time is set to be \(t_{p}\). On the other hand, if the node starts from the moving state \(p_{move}\), it will select the node speed range of \(v\in[v_{\max}-v_{\min}]\). Recall that, \(p_{pause}\) is the probability of a mobile node in the pause state, while \(t_{p}\) is the time spent by the node during pause state. The mobility of nodes in a distributed dynamic network is calculated as follows:  
\[\text{Mobility}=\min_{v\in[v_{\max}-v_{\min}]}\sum_{(s,d)\in N}f_{(s,d)}(v) \tag{8}\]  
The stability of an established path between the source and destination nodes depends on the mobility of the intermediate nodes which form the path. Therefore, a path will be unstable if the mobility of the constituent nodes of the path is high. Based on the above mobility value of nodes in the network, the MRLAM routing scheme prioritizes nodes with less mobility, meaning that the route's stability and network's lifetime are increased, resulting in reduced probability of link failure.