# The Availability of Space Service for Inter-Satellite Links in Navigation Constellations
## 4 Numerical Simulation
### None of the Satellites Are Scheduled (Ns)  
From time to time, there are essential and real-time activities using the ISLs that cannot be interrupted. Consequently, in these cases, the users only receive service sporadically, as shown in Figure 12.  
Figure 11: The available probability of SSM (the worst user): (**a**) one or more (12 satellites); (**b**) four or more (12 satellites); (**c**) one or more (six satellites); and (**d**) four or more (six satellites). Figure 12: Geometry of users receive the signal by chance when ISLs are in normal work. In the NS case, the availability will be reduced significantly and is related to the beamwidth. Considering the visibility, users should be covered in the beam of the ISL and should not be obstructed by the Earth, that is:  
\[\left\{ \begin{array}{l}\theta_{obs}<\theta_{ss}<\theta_{scan}\\ \theta_{obs}<\theta^{\prime}_{ss}<\theta_{scan}\\ \theta_{obs}<\theta_{su}\\ 0\leq\theta_{ssu}\leq\theta_{beam}/2\end{array} \right. \tag{5}\]  
where \(\theta_{ss}\) is the angle formed by the LOS of the two navigation satellite and the line from the transmitting satellite to the center of the Earth, \(\theta^{\prime}_{ss}\) is the angle formed by the LOS of the two navigation satellite and the line from the receiving satellite to the center of the Earth, \(\theta_{su}\) is the angle formed by the LOS of the transmitting satellite to the user and the line from the transmitting satellite to the center of the Earth, and \(\theta_{ssu}\) is the angle formed by the LOS of the two navigation satellite and the LOS of the transmitting satellite to the user. The first two conditions mean the two satellites can view each other so that a link can be established. The last two conditions guarantee the user is covered by the beam and is not obstructed by Earth. If the position of the transmitting satellite in ECEF coordinates is \(\overrightarrow{R_{st}}=(x_{st},y_{st},z_{st})\), the position of the receiving satellite is \(\overrightarrow{R_{sr}}=(x_{sr},y_{sr},z_{sr})\), the position of the user is \(\overrightarrow{R_{u}}=(x_{u},y_{u},z_{u})\), then the angles in Equation (5) can be computed as Equation (6) according to Equation (2):  
\[\left\{ \begin{array}{l}\theta_{ss}=\arccos(\frac{x_{st}(x_{st}-x_{sr})+y_{st }(y_{st}-y_{sr})+z_{st}(z_{st}-z_{sr})}{\sqrt{x_{st}^{2}+y_{st}^{2}+z_{st}^{2}} \sqrt{(x_{st}-x_{sr})^{2}+(y_{st}-y_{sr})^{2}+(z_{st}-z_{sr})^{2}}})\\ \theta^{\prime}_{ss}=\arccos(\frac{x_{st}(x_{st}-x_{st})+y_{st}(y_{st}-y_{st})+ z_{st}(z_{st}-z_{st})}{\sqrt{x_{st}^{2}+y_{st}^{2}+z_{st}^{2}}\sqrt{(x_{st}-x_{st})^{2}+(y_{st}-y_{st})^{2}+( z_{st}-z_{st})^{2}}})\\ \theta_{su}=\arccos(\frac{x_{st}(x_{st}-x_{su})+y_{st}(y_{st}-y_{su})+z_{st}(z_ {st}-z_{su})}{\sqrt{x_{st}^{2}+y_{st}^{2}+z_{st}^{2}}\sqrt{(x_{st}-x_{su})^{2} +(y_{st}-y_{su})^{2}+(z_{st}-z_{su})^{2}}+(z_{st}-z_{su})^{2}})\\ \theta_{ssu}=\arccos(\frac{(x_{st}-x_{su})(x_{st}-x_{sr})+(y_{st}-y_{su})(y_{st }-y_{st})+(z_{st}-z_{su})(z_{st}-z_{sr})}{\sqrt{(x_{st}-x_{sn})^{2}+(y_{st}-y_{ su})^{2}+(z_{st}-z_{su})^{2}}+(z_{st}-z_{sr})^{2}})\end{array} \right. \tag{6}\]  
The grid used in the simulation needs to be changed to \(\theta_{beam}/2\times\theta_{beam}/2\) when the satellites are unscheduled. The timing and link are decided by the link plan, which is generated based on the visibilities and requirements. There are a maximum of 12 pairs of simultaneous links between 24 satellites. There are always four visible links in the same orbit and in the different orbits for each navigation satellite. Three kinds of link plans are analyzed in this paper: all links are in the same orbit, all links are in different orbits, and half are in the same orbit. The link plans are shown in Table 2, where 11-13 means the link of MEO11 and MEO13, etc. Figure 13 demonstrates the availability of global average as a function of the altitude and link plan when \(\theta_{beam}=12^{\circ}\).