# An Efficient Routing Protocol with Overload Control for Group Mobility in Delay-Tolerant Networking
## 2 A Proposed Group Mobility Model and Group Delivery Predictability Model  
### A Proposed Group Mobility Model  
The group mobility model of this paper was extended from the RPGM model in [12], as shown in Figure 1. Each group consists of a number of mobile nodes distributed within group range from a logical group center. The next position of a logical group center is determined based on a random waypoint mobility model, and nodes within each group move based on a cluster mobility model [23]. For more elaborate modeling of the movement of nodes in a group, we placed a number of group intermediate points separated with fixed group movement distance along the straight line between two consecutive group random waypoints. If the distance between intermediate point and next random waypoint is less than the group movement distance, the next random waypoint is defined as the next intermediate point.  
Figure 2 shows a detailed movement model of nodes in a group between two consecutive group intermediate points between two consecutive random waypoints of a logical group center. The notations in Figure 2 are defined as follows, which are partly based on [13]:  
* Position of a group center \(j\) at time \(t\), based on random waypoint mobility model: \(\mathbf{Y_{j}}(t)\)
* Velocity of a group center \(j\) at time \(t\): \(\mathbf{W_{j}}(t)\)
* Position of an intermediate point of a group center \(j\) at time \(t\): \(\mathbf{I_{j}}(t)\)
* Group movement distance, i.e., the distance between two consecutive intermediate points of a group center: \(GD\)
* Position of an intermediate point of a group center \(j\) at time \(t_{m}+t_{k+1}\): \(\mathbf{I_{j}}(t_{m}+t_{k+1})=\mathbf{I_{j}}(t_{m}+t_{k})+GD\times\left(\mathbf{Y_{j}}(t_ {m+1})-\mathbf{Y_{j}}(t_{m})\right)/\left|\mathbf{Y_{j}}(t_{m+1})-\mathbf{Y_{j}}(t_{m})\right|\)
* Position of a node \(i\) in a group \(j\) at time \(t\): \(\mathbf{X_{j,i}}(t)\)
* Local displacement position of a node \(i\) in a group \(j\) at time \(t\) based on cluster mobility model: \(\mathbf{Z_{j,i}}(t)\)
* Next position of a node \(i\) in a group \(j\) at time \(t_{m}+t_{k+1}\): \(\mathbf{X_{j,i}}(t_{m}+t_{k+1})=\mathbf{X_{j,i}}(t_{m}+t_{k})+\mathbf{Z_{j,i}}(t_{m}+t_{k })+\left(\mathbf{I_{j}}(t_{m}+t_{k+1})-\mathbf{I_{j}}(t_{m}+t_{k})\right)\)  
In Figure 2, a group center is at \(\mathbf{I_{j}}(t_{m}+t_{k})\) and \(\mathbf{I_{j}}(t_{m}+t_{k+1})\) when \(t=t_{m}+t_{k}\) and \(t=t_{m}+t_{k+1}\), respectively. A node \(i\) in a group \(j\) was at \(\mathbf{X_{j,i}}(t_{m}+t_{k})\) when \(t=t_{m}+t_{k}\). The next  
Figure 1: A considered group mobility model.  
Figure 2: A movement model of nodes in a group.  
position of a node \(i\) in a group \(j\) at \(t=t_{m}+t_{k+1}\), \(\mathbf{X}_{ji}(t_{m}+t_{k+1})\), is determined by adding the local displacement position based on cluster mobility model, \(\mathbf{Z}_{ji}(t_{m}+t_{k})\), and the difference between two consecutive group intermediate points, \(\mathbf{I}_{j}(t_{m}+t_{k+1})-\mathbf{I}_{j}(t_{m}+t_{k})\). Finally, the velocity of a node \(i\) in a group \(j\) at \(t=t_{m}+t_{k+1}\) is determined as follows:  
* Velocity of a node \(i\) in a group \(j\) at time \(t_{m}+t_{k+1}\): \(\mathbf{V}_{ji}(t_{m}+t_{k+1})=\mathbf{W}_{j}(t_{m})\times\big{(}\big{|} \mathbf{X}_{ji}(t_{m}+t_{k+1})-\mathbf{X}_{ji}(t_{m}+t_{k})\big{|}/\big{|} \mathbf{I}_{j}(t_{m}+t_{k+1})-\mathbf{I}_{j}(t_{m}+t_{k})\big{|}\big{)}\)