## 2 Problem Statement
### Significant Burden in Computational Complexity  
The number of fault subsets should satisfy that the actual monitored fault probabilities are not lower than the required monitored fault probability. The relationship can be given as the following inequality:  
\[P_{fault,actual}>P_{fault,req} \tag{3}\]  
where the actual monitored fault probabilities \(P_{fault,actual}\) and the required monitored fault probabilities \(P_{fault,req}\) is calculated as:  
\[P_{fault,actual}=\sum_{k=1}^{N_{faultmodes}}p_{fault,k}=\sum_{k=1}^{N_{ faultmodes}}\left(\prod_{i\in idx}(P_{event,i})\prod_{j\
otin idx}\left(1-P_{ event,j}\right)\right) \tag{4}\]  
\[P_{fault,req}=1-p_{no\_fault}-P_{THRES} \tag{5}\]  
where \(p_{fault,k}\) refers to the probability of \(k\)th fault mode. \(P_{THRES}\) is the threshold for the integrity risk coming from unmonitored faults. \(N_{faultmodes}\) is the total number of fault modes. \(p_{no\_fault}\) is the probability of all-in-view subset, which means no fault exists:  
\[p_{no\_fault}=\prod_{k=1}^{(Nsat+Nconst)}\left(1-P_{event,k}\right) \tag{6}\]  
where (\(N_{sat}+N_{const}\)) represents the total number of fault events.  
According to Equation (6), it is easy to understand that with more constellations involved in ARAIM, the value of \(p_{no\_fault}\) will get smaller as more fault events are involved. It will result in a bigger \(P_{fault,req}\) according to Equation (5). Meanwhile, on the other side of inequality (3), the probability of original \(k\)th fault mode gets smaller simultaneously. Figures 1 and 2 show the probability trends of two fault scenarios: One satellite fault mode and total required probability for monitor, calculated by (2) and (6), respectively. The \(P_{THRES}\), \(P_{const,j}\) and \(P_{sat,j}\) are set as \(8\times 10^{-8}\), \(1\times 10^{-4}\), and \(1\times 10^{-5}\), respectively.  
One satellite fault is the basic fault mode in ARAIM. As shown in Figure 1, the probability of one satellite fault decreases with the increase in visible satellites. What is worse, more constellations result in smaller probability. Similar probability loss can be derived in the other fault modes, like two satellites fault and one satellite and one constellation fault. On the other hand, as shown in Figure 2, with the increase in visible satellites, the total required probability for monitor increases linearly. More constellations result in higher required probability. As one falls, another rises. The result is that more fault modes are needed to be monitored and more subsets are needed to be calculated to meet the inequality (3).  
The recent studies unexceptionally verified the exponential increase in fault subsets. The total number that need to be monitored can be hundreds and thousands. When the number of simultaneous faults exceeds two, another core question refers to whether it needs to categorize the multiple satellites fault to the constellation fault. This is outside the scope of this article. We do not consider the specific number, but inevitably the explosive fault modes are significant computational burdens for the airborne receiver.  
There are two more potential hazards in the computation burden. The first is the integrity after fault exclusion. Once one fault is detected and the corresponding satellite(s) are excluded, the solution separation will be executed again and every new subset will calculate the fault-tolerant positioning results, which means heavy computation load. The second one is the uncertainty of constellation and satellite prior fault probability. There's no definite conclusion about the order of magnitude of fault probabilities. In recent studies, \(10^{-5}\), \(10^{-4}\), and \(10^{-3}\) are all considered [29]. However, it has a great impact on the number  
Figure 1: Probability of one satellite fault.  
Figure 2: Required probability for monitor.  
of subsets. The current research has shown that the bigger of the prior satellite fault probability is, the more the number of subsets. Particularly, the increase can be an order of magnitude.