## 3 Preliminary Information
### Random Waypoint Mobility Model  
**Definition 3**. : _In the Random Waypoint Mobility (RWP) model, the migration process of each node is as follows: randomly select a waypoint \(X\) in the network area; randomly select a value which is in the interval \([v_{\min},\,v_{\max}]\) as movement speed \(v\); the node moves to waypoint \(X\) with speed \(v\); stops in the waypoint \(X\) for a random duration \(T_{\text{stop}}\) until the epoch is over, where \(T_{\text{stop}}\) is randomly distributed in the interval \([0,\,T_{\max}]\); repeat the above migration process._  
**Lemma 2**. : _If the mobile area of a node is a rectangle, then the Random Waypoint Mobility Model is characterized by the following:_  
_In one epoch, the mean movement distance for a node is:_  
\[\overline{L}\approx 0.5214\sqrt{D}. \tag{2}\]  
_The probability that a node locates in location (x,y) is unevenly distributed, and the probability density function is:_  
\[f(x,y)\approx\frac{36}{D^{3}}(x^{2}-\frac{D}{4})(y^{2}-\frac{D}{4}). \tag{3}\]  
_In one epoch, the mean movement duration is:_  
\[\overline{T}=\overline{L}/\overline{v}. \tag{4}\]  
_The probability of the movement direction of a node doesn't obey a uniform distribution. The direction is pointed to the center of the network with high probability. If the center of the network is the origin, then the probability density function of movement direction \(\theta\) is:_  
\[f(\theta)=\frac{1}{4\left|\sin^{3}\!\theta\right|}(|\sin\!\theta|\times g( \theta)+\arcsin(|\sin\!\theta|)\times\cos\!\theta) \tag{5}\]  
_where \(g(\theta)=-2\text{cos}^{4}\theta-2\text{cos}^{3}\theta\times|\text{cos}\theta |+\text{cos}^{3}\times|\text{cos}\theta|+\text{cos}^{2}\theta+\text{cos}\theta \times|\text{cos}\theta|+1\)._  
The location of each node is likely biased towards the center of the network with the movement of the node in the Random Waypoint Mobility model, so this kind of non-uniform distribution of nodes makes it difficult to analyze the Random Waypoint Mobility Model. **Lemma 3**. : _In the Random Waypoint Mobility model, the average hitting probability of a node in an epoch is \(\overline{P}_{rw}^{H}\)_  
\[\overline{P}_{rw}^{H}=\frac{2R\times\overline{L}}{D} \tag{6}\]  
**Proof. ** Node \(A\) moves in the Random Waypoint Mobility Model, node \(B\) is static in a randomly selected location \(X_{B}=(x,y)\) in the network. Further, we suppose that node \(A\) locates at the location \(X_{s}\) at the beginning of some epoch, and locates at location \(X_{f}\) at the end of the epoch, node \(X_{B}^{\perp}\) in the line between the node \(X_{s}\) and the node \(X_{f}\) is the node nearest to the node \(X_{B}\). When and only when \(\parallel X_{B}-X_{B}^{\perp}\parallel\leq R\), node \(A\) hits node \(B\) in the current epoch. Suppose all possible collection of epochs can be denoted as \(z=\left\{(X_{s},X_{f}):X_{s},X_{f}\in U\right\}\), then all collection of epochs in which node \(B\) would be hit can be denoted as \(z_{hit|B}=\left\{(X_{s},X_{f}):\parallel X_{B},X_{B}^{\perp}\parallel\leq R\right\}\), so the probability that node \(A\) hits node \(B\) can be calculated by the formula as follow:\[P_{rw}^{H}(x,y)=\frac{\parallel z_{hit|\text{B}}\parallel}{\parallel Z\parallel}= \frac{\parallel z_{hit|\text{B}}\parallel}{\iint\limits_{U}dX_{S}dX_{f}}=\frac{ \parallel z_{hit|\text{B}}\parallel}{D^{2}}. \tag{7}\]  
Analogously, the set of all epochs in which node \(A\) is passed through location \(X_{B}\) can be denoted as \(z_{hit|\text{B}}^{*}\left\{(X_{s},X_{f}):\parallel X_{B}-X_{B}^{\perp} \parallel\leq\delta,\delta\to 0\right\}\). The proportion of this kind of epoch to total is \(\frac{\parallel z_{hit|\text{B}}^{*}\parallel}{D_{2}}\) neglecting the boundary influence.