## 3 Preliminary Information  
In mobile wireless networks, node mobility affects the quality of the wireless channel. The dynamic changes of the links between nodes make it harder to design routing protocols [31]. Mobility-assisted protocols based on encounters are put forward to alleviate the situation. The main idea behind the encounter-based protocols is that as a sensor node wants to relay message, it is not required to look for a link to relay the message immediately but to retain the message until the node meets the message recipients or the nodes which enable forwarding the message to the recipients, and then the node forwards the message. This kind of message forwarding way dependent on an encounter-based protocol does not need the overhead of maintaining traditional multi-hop routing paths, and it has been widely used in delay tolerant networks. Research on the statistical characteristics of node meeting, such as the expected meeting time, mean delay time and so on, is of great significance in improving the performance of the protocol due to messaging when only when related nodes meet. Some relevant symbols used in the mobile model definition are given in Table 1.  
There are two definitions given as follows:  
**Definition 1**.: _If node \(i\) is mobile, whose movement mode is \(mm\), and node \(j\) is static, then the hitting probability is the probability that node \(i\) hits node \(j\) within the time \(T\), which is represented as \(P^{H}_{mm}(t)\), and \(P^{H}_{mm}(t)=P(\exists\tau\in(0,t]:\parallel X_{i}(\tau)-X_{j}\parallel<R)\)._  
**Definition 2**.: _If node \(i\) and node \(j\) are both mobile, their movement mode are \(mm\), then meeting probability is the probability that node \(i\) encounters node \(j\) within the time \(T\), which is expressed as \(P^{M}_{mm}(t)\), \(P^{H}_{mm}(t)=P(\exists\tau\in(0,t):\parallel X_{i}(\tau)-X_{j}(\tau)\parallel <R)\)._  
**Lemma 1**.: _In the movement mode based on epoch, if the speed of node is randomly generated in the interval \([v_{\min},v_{\max}]\), where \(v_{\min}>0,v_{\max}<\infty\), then the mean speed of the node satisfies:_  
\[\overline{v}=\frac{v_{\max}-v_{\min}}{\ln\frac{v_{\max}}{v_{\min}}} \tag{1}\]  
\begin{table}
\begin{tabular}{p{42.7 pt} p{341.4 pt}} \hline \hline
**Symbol** & **Denotation** \\ \hline \(R\) & Transmission radius of node. \\ \hline \(X_{i}(t)\) & Position of node \(i\) at time \(t\). \\ \hline Epoch & The procedure during which a node moves to somewhere at the same rate, and in the same direction, and then stops for a while. \\ \hline \(L\) & The length of an epoch, which is the distance from the point of epoch beginning to the point of epoch ending. \\ \hline \(v\) & The speed of node in an epoch, which is distributed in the interval \([v_{\min},v_{\max}]\), \(\overline{v}\) is the mean speed. \\ \hline \(T_{stop}\) & The random residence time selected by node when an epoch is over, the length of residence time is randomly distributed in the interval \([0,T_{\max}]\), \(\overline{T}_{stop}\) is the mean value of \(T_{stop}\) \\ \hline \(\overline{T}\) & The average length of time that node spends in movement status before an epoch is over. \(\overline{T}=\frac{\overline{T}}{\overline{V}}\), the total time is \(\overline{T}+\overline{T}_{stop}\). \\ \hline \(v_{mm}\) & The relative speed between node \(i\) and node \(j\) when they move in movement mode mm, \(v_{mm}=\left|\overline{v}_{i}-\overline{v}_{j}\right|\), where \(\overline{v}i\) is the velocity vector of the node \(i\). \\ \hline \(\hat{v}_{mm}\) & The standard relative speed of movement mode mm. \(\hat{v}_{mm}=\frac{\overline{v}_{mm}}{\overline{V}}\). \\ \hline \end{tabular}
\end{table}
Table 1: Symbols and notations.