# BDS-3 Integrity Risk Modeling and Probability Evaluation
## 3 BDS-3 Integrity Failure Model and Effect Analysis Algorithm
### BDS-3 Integrity Risk Tree Modeling
#### 3.4.2 Integrity Risk Probability Evaluation Method
In the formula, \(k\) is the number of historical occurrences of abnormal events, \(T\) is the statistical time (in hours), and \(MTTN\) indicates the Mean Time To Notify, which represents the average time used to identify abnormal events. At present, the average abnormal response time promised by BDS-3 in ICAO SARPs is 1 h.  
However, due to the short service time of BDS-3 and the influence of factors such as the system's ground state update and onboard key stand-alone equipment aging, the number of system abnormal events varies with time. If Equation (4) is directly adopted, the time-varying characteristics of abnormal event occurrence are considered, and the estimation redundancy of the number of abnormal event occurrences cannot be given. Therefore, it is next to impossible to give a reasonable evaluation result of the occurrence probability of abnormal events with Equation (4). To more accurately evaluate the abnormal event occurrence probability, it is necessary to establish a prediction model for the number of abnormal event occurrences. The statistical data of historical abnormal events are used as input to establish the functional mapping relationship between the statistical number of historical abnormal events and the abnormal event occurrence probability and realize the reasonable prediction of the abnormal event occurrence probability in the risk tree. Considering that the BDS-3 satellite has the function of software on-orbit reconstruction, in the actual operation and maintenance process, the relevant on-orbit reconstruction work is implemented on demand from time to time, according to the overall engineering requirements, technical status upgrades, software robustness upgrades and other inputs. Therefore, for BDS-3, when predicting the future abnormal probability based on historical abnormal data, the following assumptions should be considered:  
1. During the estimation period, satellites are not reconstructed on-orbit software, i.e., the hardware and software states that cause anomalies are considered relevant and consistent. 2. In the next period of time, the hardware related to the space segment and the ground control segment does not run in a super-life state. 3. Assume that the system is a stable system, that is, the fault occurrence probability is a constant within a certain time, and an abnormal occurrence is a rare event. Based on the above assumptions, this paper uses the Bayesian estimation theory to calculate the abnormal events probability in the system, as follows:  
\[f(r|k)=\frac{P(k|r)f(r)}{\int P(k|r)f(r)dr} \tag{5}\]  
In Equation (5), \(k\) is the number of abnormal events obtained by statistics at time \(T\) and belongs to posterior information. \(r\) is the expected occurrence number of abnormal events and belongs to prior information. The main function of the Bayesian estimation theory is to use posterior information to infer prior information. _P(k/r)_ is the probability that the number of abnormal events occurrences is \(k\) during time \(T\), under the condition that the occurrence probability of abnormal events is \(r\). According to the operation and maintenance experience and literature [10; 12; 13], the Poisson distribution can be used to describe:  
\[P(k|r)=\frac{r^{k}}{k!}e^{-r} \tag{6}\]  
_f(r)_ is the continuous prior probability density function for \(r\), which is the occurrence probability of an abnormal event, usually described by a gamma distribution:  
\[f(r)=\frac{\partial^{r}r^{-1}e^{-\partial r}}{\Gamma(v)} \tag{7}\]  
In Equation (7), \(\partial\) is the shape parameter, \(v\) is the scale parameter, and \(\Gamma()\) is the gamma distribution function. Substituting Equations (6) and (7) into Equation (5), we can obtain that:\[f(r|k,\partial,v)=\frac{(1+\partial)^{k+v}r^{k+v-1}e^{-(1+\partial)r}}{\Gamma(k+v)} \tag{8}\]  
Although the gamma distribution can be used to describe the prior probability density function, the values of \(\partial\) and \(v\) need to be determined according to the actual situation. [PERSON] and [PERSON] and [PERSON] et al.