## 3 Improved Multi Mobile Agent Data Fusion Scheme
### Link Stability in Mobile Environments
For a sensor node moving at a speed of \(v_{1}\), the cumulative distribution function (CDF) of the link lifetime is:  
\[ \begin{array}{c}F_{link}^{v1}=1-\frac{1}{\pi(b-a)}\\ \int_{0}^{\pi}\int_{0}^{2\frac{R}{2}}\frac{v}{\sqrt{v^{2}+v1^{2}+2 vv1\cos\varpi }}\sqrt{1-\left(\frac{gt}{2R}\right)^{2}}\\ \times[u\Big{(}\sqrt{v^{2}+v1^{2}+2 vv1\cos\varpi}-a\Big{)}\\ -u\Big{(}\sqrt{v^{2}+v1^{2}+2 vv1\cos\varpi}-b\Big{)}]dvd\varpi\end{array}  \tag{34}\]  
For Equation (34) on the \(T\) differential, one can get link life proposed distribution function (PDF) \(f_{link}^{v_{1}}(t)\). It is important that network nodes and location distribution do not influence the expression. In Figure 11, the sensor node 1 is moving at a speed of V relative to the previous static coordinate system XY, and the total expected value of the number of nodes entering the shadow area in \(t\) seconds is:  
\[ \begin{array}{c}E(\text{the number of nodes that will enter the area in $T$ seconds})=\eta(v)\\ =\frac{2R\varepsilon l}{\pi(b-a)}\{b^{2}\varepsilon\big{(}\frac{v}{b}\big{)}-2 a^{2}\varepsilon\big{(}\frac{v}{a}\big{)}+a^{2}\varepsilon\big{[}\pi-sin^{-1} \big{(}\frac{a}{b}\big{)},\frac{v}{a}\big{]}\\ \qquad+\frac{v^{2}}{4}\int_{o}^{\pi}[1+\cos(2\phi)]\lg\Big{|}\frac{b+\sqrt{b^{ 2}-v^{2}sin^{2}\phi}}{v+v\cos\phi}\Big{|}d\phi\\ -\frac{v^{2}}{4}\int_{\pi-\sin^{-1}a/v}^{\pi}[1+3\cos(2\phi)]\lg|\frac{a+\sqrt{ a^{2}-v^{2}sin^{2}\phi}}{a-\sqrt{a^{2}-v^{2}sin^{2}\phi}}|d\phi\}\end{array}  \tag{35}\]  
where \(\varepsilon\{\bullet\}\) is a complete elliptic integral, \(\varepsilon\{\bullet,\bullet\}\) is a non- complete elliptic integral, and \(\Phi\) is the angle between the direction of movement and the level. Therefore, when the minimum value of the rate is 0, the expected number of nodes per second into the transfer area is simplified to:  
\[\begin{array}{c}\overline{\eta}(v)=\frac{2R\varepsilon l}{\pi b}\{b^{2} \varepsilon\big{(}\frac{v}{b}\big{)} \tag{36}\]  
where \(\sigma=\rho/\pi R^{2}\) node / m\({}^{2}\), \(\rho\) stands for the average node in a transmission region. The new link to node expected arrival rate is proportional to the average density of nodes in the network transmission and proportional to the radius \(R\) of the node. In the above Figure 11, for a given \(v\), the probability that cross link arrival time is greater than \(t\) is equal to the probability of the presence of at least one rate for \(v\) nodes in the shaded area. Thus, the cumulative distribution function of new cross link arrival time can is:  
\[ \begin{array}{c}F_{arrival}^{v_{1}}(t)=1-\frac{1}{\pi(b-a)}\\ \int_{0}^{\pi}\int_{0}^{\infty}e^{-2R\varepsilon lv}\frac{v}{\sqrt{v^{2}+v_{1} ^{2}+2 vv_{1}\cos\varpi}}\\ \ast[u(\sqrt{v^{2}+v_{1}^{2}+2 vv_{1}\cos\varpi}-a)\\ -u(\sqrt{v^{2}+v_{1}^{2}+2 vv_{1}\cos\varpi}-b)]dvd\varpi\end{array}  \tag{37}\]  
New link cross arrival time for PDF \(f_{arrival}^{v_{1}}(t)\) is:  
\[ \begin{array}{c}f_{arrival}^{v_{1}}=\frac{d}{dt}F_{arrival}^{v_{1}}(t)=\frac {2R\sigma}{\pi(b-a)}\\ \int_{0}^{\infty}\int_{0}^{\infty}e^{-2R\varepsilon lv}\frac{v}{\sqrt{v^{2}+v_ {1}^{2}+2 vv_{1}\cos\varpi}}\\ \ast[u(\sqrt{v^{2}+v_{1}^{2}+2 vv_{1}\cos\varpi}-a)\\ -u(\sqrt{v^{2}+v_{1}^{2}+2 vv_{1}\cos\varpi}-b)]dvd\varpi\end{array}  \tag{38}\]We infer that the PDF curve of the new cross arrival time decreases rapidly with the increase of time \(t\). Using the Figure 12, we calculate the distribution of the interrupt arrival time of the link. The link interrupt cross arrival time for CDF is:  
\[ \begin{array}{l}F_{break}^{v_{1}}(t)=P\{\text{Arrival time of link interrupt crossing}\leq t\}\\ =\iint(1-e-2R\sigma{}lv)f(v,\phi)dvd\phi\\ \\ =1-\frac{1}{\pi(b-a)}\\ \int_{0}^{\pi}\int_{0}^{\infty}e^{-2R\sigma{}lv}\frac{v}{\sqrt{v^{2}+{v_{0}} _{1}^{2}+2 vv_{1}\cos\varpi}}\\ \ast[u(\sqrt{v^{2}+{v_{1}}^{2}+2 vv_{1}\cos\varpi}-a)\\ -u(\sqrt{v^{2}+{v_{1}}^{2}+2 vv_{1}\cos\varpi}-b)]dvd\varpi\end{array}  \tag{39}\]  
The distribution of the link interrupt arrival time and the cross-arrival time of the new link is the same.