## 3 Link Cost Metric With Reliability Assurance
### Reliability Model for Network With EO Missions  
In SCNs, the shared navigation information has some measuring errors, which cause the deviations between actual values of satellite position and the measured values. Additionally, this also leads to the differences that some essential data are calculated by math recursion on the orbit dynamic model. The deviations and differences create the state of wireless links in some uncertain situations, such as link interruption, lower bandwidth, etc. In this paper, it is assumed that the probability of every working link in normal work is \(r(0<r<1)\) and the effective working time per period for each satellite is \(\tau\). Then, each link's reliability model can be represented as:  
\[Y=r\tau T^{-1} \tag{8}\]  
which represents the probability of normal work per period. According to the proposed cooperative work  
In any EO mission, it can be regarded that the state of the network operating normally means all image data produced from EOS can be transmitted to GSC successfully and the construction of each sub-network \(s_{i}\rightarrow\left(t_{1},t_{2},\cdots,t_{m}\right)\) of EOS to GSC are independent events. According to the independence of probability theory, let the probability of the normal work state be \(P\), and it meets the following relations:  
\[ \begin{array}{rcl}P\left\{\left(s_{1},s_{2},\cdots,s_{n}\right)\rightarrow \left(t_{1},t_{2},\cdots,t_{m}\right)\right\}&=&P\left\{\left(s_{1}\rightarrow \left(t_{1},t_{2},\cdots,t_{m}\right)\right)\bigcap\cdots\bigcap\left(s_{n} \rightarrow\left(t_{1},t_{2},\cdots,t_{m}\right)\right)\right\}\\ &=&P\left(s_{1}\rightarrow\left(t_{1},t_{2},\cdots,t_{m}\right)\right) \bigcap\cdots\bigcap P\left(s_{n}\rightarrow\left(t_{1},t_{2},\cdots,t_{m} \right)\right)\\ &=&\prod\limits_{i=1}^{n}P_{i}\end{array}  \tag{11}\]  
Thus, combined with the equation above, to guarantee stable image data transmission, on behalf of the independence of \(P_{i}\), network reliability meets the constraints below:  
\[\max P=\max\prod\limits_{i=1}^{n}P_{i}=\prod\limits_{i=1}^{n}\max P_{i} \tag{12}\]  
Equation (12) shows that the optimal reliability of the whole network topology requires each sub-topology's reliability to be optimal. Additionally, it is obvious that when the sub-topology is designed as the configuration in Figure 5, i.e., all paths are in parallel, the reliability is the best. Then, Equation (11) can be changed as follows:  
\[ \begin{array}{rcl}\max P&=&\prod\limits_{i=1}^{n}\max\left[1-\prod\limits_{ j=1}^{m}\left(1-Y_{ij}\right)\prod\limits_{q=1}^{k}\left(1-Y_{iq}\right)\right] \\ &=&\prod\limits_{i=1}^{n}\max\left\{1-\prod\limits_{j=1}^{m}\left[1-r^{\alpha_ {ij}}\prod\limits_{e=1}^{\alpha_{ij}}\left(\tau_{e}T_{e}^{-1}\right)\right] \prod\limits_{q=1}^{k}\left[1-r^{\alpha_{iq}}\prod\limits_{e=1}^{\alpha_{iq}} \left(\tau_{e}T_{e}^{-1}\right)\right]\right\}\end{array}  \tag{13}\]  
s.t. :  
\[ \begin{array}{rcl}\left\{1\leq a_{ij}\leq N-1\right.\\ 1\leq a_{iq}\leq N-1\\ 1\leq Y_{j},Y_{q},r\leq 1\end{array}  \tag{14}\]  
where, \(k\) is the number of redundant paths of the sub-topology \(s_{i}\rightarrow\left(t_{1},t_{2},\cdots,t_{m}\right)\). In Equation (14), when \(a_{j}=1\), \(a_{k}=2\), it can be proved that the network has the best reliability assurance and the probability of normal work is \(\prod\limits_{i=1}^{n}\left[1-\prod\limits_{j=1}^{m}\left(1-r\tau_{j}T_{j}^{-1 }\right)\prod\limits_{q=1}^{k}\left(1-r\tau_{q}T_{q}^{-1}\right)\right]\). However, under the actual work conditions, limited by the maximum communication range, it is difficult to make \(a_{j}=1\), i.e., transmitting data from EOS to GSC directly. Figure 5: Topology configuration with the highest reliability (\(m\) shortest paths, \(k\) redundant paths). Furthermore, because all of the variables in Equation (13) are not correlative, it can be further simplified as:  
\[\max P=\left\{ \begin{array}{l}\max k\\ \min\alpha_{ij}\\ \min\alpha_{iq}\end{array} \right.