# Uncertainty Theory-Based Resilience Analysis for LEO Satellite Communication Systems
## 4 LEO-SCS and Its Uncertain Evolution Model
### Uncertain Satellite Network Evolution Model
#### 4.2.4 Evolution Rule of Dynamic Uncertain Satellite Network Model
When the second failed satellite was recovered using its in-orbit backup at time \(t=221\) h, the system availability increased to the level under the single satellite failure. At time \(t=622\) h, the first failed satellite was recovered using the ground backup, and the system availability returned to the normal. In the case above, the backup strategies included both in-orbit and ground backup. When dual satellite failures occur, the possibility that different recovery strategies are adopted, as shown in Table 5. The number of in-orbit backups in each orbit is 1. One can see that over 90% of dual satellite failures use at least one ground backup. Ground backup takes longer to restore the system, and during this time, other satellites are more likely to fail. Therefore, the probability of dual satellite failures can be reduced by increasing the number of in-orbit backups and decreasing the transfer time of the ground backup. Using this method, system resilience can be improved. According to the analysis above, the satellite recovery time significantly affects the system belief instantaneous availability and thus the system resilience. In an LEO-SCS,

Figure 8: System belief instantaneous availability under dual satellite failures. Figure 7: System belief instantaneous availability when two satellites in the same orbit fail successively. the number of in-orbit backups and the satellite orbital lifetime determine the recovery time in practice. We calculated the system resilience under different satellite compositions and backup strategies when the requirement for the system's behavior after a disruption was 0.99. The results are shown in Table 6.

As shown in Table 6, adding more in-orbit backups and extending satellite orbital lifetimes can improve the resilience of an LEO-SCS because more in-orbit backup reduces satellite recovery time and a longer satellite orbital lifetime reduces the probability of satellite failure. The initial increment on in-orbit backups is more effective than subsequent ones. For example, when the number of in-orbit backups increases from 1 to 2, the system resilience increases more than when backups increase from 2 to 3. Therefore, when designing an in-orbit backup strategy for an LEO-SCS, it is necessary to consider satellite orbital lifetimes, system resilience, and cost comprehensively. ## 7 Conclusions

We proposed a belief resilience measure. Based on this, a dynamic uncertain satellite network evolution model was developed to describe a dynamic uncertain LEO-SCS, and a methodology was proposed to estimate the resilience of the model. A Walker LEO-SCS with 48 satellites was used to demonstrate the effectiveness of our method, and the influence of the number of in-orbit backups and the satellite orbital lifetime on system resilience was analyzed. The results show that the proposed measure can be used to quantify the resilience of systems with aleatory and epistemic uncertainties. The recovery time of satellites is the main factor affecting system resilience, and the number of in-orbit backup satellites and the satellite orbital lifetime play important roles in increasing LEO-SCS resilience. The proposed resilience evaluation methodology can be used for LEO-SCS comparison, optimization, and design. The number of satellites in an LEO-SCS has reached hundreds or thousands. How to quickly and accurately assess the resilience of such giant systems will be studied further in our future research. Conceptualization, J.M. and R.K.; methodology, J.M. ; software, J.M. ; validation, J.M. and R.L. ; formal analysis, J.M., R.L. and Q.Z. ; investigation, J.M. and R.L. ; resources, L.L. and X.W. ; writing--original draft preparation, J.M. ; writing--review and editing, J.M., R.L., R.K. and Q.Z. ; supervision, R.L. and R.K.; project administration, R.L. and R.K. All authors have read and agreed to the published version of the manuscript.