# Through-Life Maintenance Cost of Digital Avionics
## 7 Results and Discussion
### Warranty Maintenance  
Figure 1 shows the dependency of MTBUR on the in-flight conditional probability of a false-positive computed by Equation (A1) when \(T=10,000\) h, \(\tau=2.5\) h, \(\lambda=10^{-4}\) h, and \(\theta=10^{-4}\) h.  
As we can see in Figure 1, the dependence of MTBUR on \(P_{flight}^{FP}\) can be conditionally divided into three intervals. On the interval from 0 to \(10^{-5}\),  
Figure 1: The dependency of MTBUR on the conditional probability of false-positive during flight. pendent of \(P_{flight}^{FP}\). When \(P_{flight}^{FP}\) changes from \(10^{-5}\) to \(10^{-4}\) MTBUR weakly depends on the conditional probability of false-positive during flight and falls by 10%. On the interval from \(10^{-4}\) to \(10^{-2}\), MTBUR strongly depends on \(P_{flight}^{FP}\) and decreases by eight times. As it follows from the analysis, the conditional probability of a false positive in-flight should be, at best, less than \(10^{-5}\), or at least no more than \(10^{-4}\) for the given data. Figure 2 shows the dependency of the in-flight conditional probability of a false-positive on the number of in-flight test checks calculated by Equation (17) when \(\alpha=10^{-6}\). We can conclude from Figure 2 that to provide the in-flight conditional probability of a false-positive in the interval \(10^{-5}\)-\(10^{-4}\), the number of in-flight test checks should be 10 to 100. Let's calculate the optimal quantity of spare LRU in the airline's warehouse with the given initial data: \(T_{W}=5000\) h, \(\tau=2.5\) h, \(q=3\), \(t_{M}^{O-level}=0.5\) h, \(\lambda=1\times 10^{-4}\) h\({}^{-1}\), \(\theta=1\times 10^{-4}\) h\({}^{-1}\), \(T_{RS}=120\) h, \(T_{ED}=40\) h, and \(t_{STOP}=1\) h.  
Figure 3 shows the dependency of the mean waiting time \(t_{spare}\) on the number of planned spare LRU (PS) for the first WM option at \(N_{W}=10\). Four spare LRU are required for \(P_{flight}^{FP}=1\times 10^{-4}\) and five spare LRU for \(P_{flight}^{FP}=1\times 10^{-3}\) to provide the mean waiting time for a spare LRU not exceeding 0.5 h.  
Figure 3: Dependences of the mean waiting time on the number of planned spare LRU (\(PS\)) for the first WM option; (**a**) circles: \(P_{flight}^{FP}=1\times 10^{-4}\), squares: \(P_{flight}^{FP}=1\times 10^{-3}\); (**b**) black circles: \(\theta=1\times 10^{-4}\) h\({}^{-1}\), red triangles: \(\theta=1\times 10^{-3}\) h\({}^{-1}\). Figure 2: The dependency of the in-flight conditional probability of a false-positive event on the number of test checks. Figure 3b shows the same dependency as in Figure 3a but with different rate of intermittent faults and \(P_{flight}^{FP}\ =\ 1\times 10^{-4}\). As we can see in Figure 3b, four spare LRU are required for \(\theta=1\times 10^{-4}\) h\({}^{-1}\) and seven spare LRU for \(\theta=1\times 10^{-3}\) h\({}^{-1}\) to provide the mean waiting time for a spare LRU not exceeding 0.5 h.  
As we can see in Figure 3a,b, the mean waiting time \(t_{spare}\) significantly depends on the conditional probability of in-flight false-positive and the rate of intermittent faults. Figure 4a shows the dependency of the optimal quantity of spare LRU on the quantity of aircraft for the WM option number one when \(P_{flight}^{FP}\ =\ 1\times 10^{-4}\) (red circles), \(P_{flight}^{FP}\ =\ 1\times 10^{-3}\) (black squares) and \(\theta=1\times 10^{-4}\) h\({}^{-1}\). Analysis of Figure 4a allows us to draw the following conclusions: the dependency of \(PS(N_{W})\) on \(N_{W}\) is an integer increasing function, a rise in the conditional probability of in-flight false-positive from \(1\times 10^{-4}\) to \(1\times 10^{-3}\) results in a substantial increment of the quantity of spare LRU (\(PS\)), and a larger increment in the quantity of spare LRU corresponds to a larger value of the \(P_{flight}^{FP}\) probability. The latter circumstance indicates that a high conditional probability of in-flight false-positive has a significant impact on the effectiveness of the first WM option.