# Through-Life Maintenance Cost of Digital Avionics
## 3 Mathematical Modeling of Avionics Maintenance
Substituting (9) to (10) gives  
\[E(TBUIR_{T})=\sum\limits_{k=0}^{M}\int\limits_{k\tau}^{(k+1)\tau}\Biggl{\{} \sum\limits_{k=1}^{k}\
u\tau P^{FP}[(\
u-1)\tau,\
u\tau|\eta][1-\Omega(\
u\tau )]+\sum\limits_{l=1}^{k}l\tau P^{TP}[(l-1)\tau,l\tau|\eta]\int\limits_{(l-1) \tau}^{l\tau}\omega(x)dx+\\ \\ (k+1)\tau P^{TP}[(k-1)\tau,k\tau|\eta][1-\Omega(k\tau)]\Bigr{\}}f(\eta)d\eta+ \int\limits_{T}^{\infty}\Biggl{\{}\sum\limits_{k=1}^{M}k\tau P^{FP}[(k-1)\tau, k\tau|\eta][1-\Omega(k\tau)]+\\ \\ \sum\limits_{k=1}^{M}k\tau P^{TP}[(k-1)\tau,k\tau|\eta]\int\limits_{(k-1)\tau}^ {k\tau}\omega(x)dx+T\times P^{TP}[(M-1)\tau,M\tau|\eta][1-\Omega(T)]\Bigr{\}} f(\eta)d\eta. \tag{11}\]  
In (11), to calculate the MTBUR, it is needful to have information about the in-flight false-positive and true-positive probabilities and the PDF of the time to permanent failure and intermittent fault. According to [30], the exponential distribution is suitable to describe the failure distribution of complicated technical equipment. Onboard electronic systems have a complex structure and include a considerable quantity of electronic elements. The influence of external and internal mechanisms on the electronic components may lead to LRU failure. Various failure mechanisms can add up, leading to a constant failure rate, which is possible only with an exponential law of distribution of the operating time to failure [31]. Let's determine MTBUR for the exponential failure and fault distribution with the following PDF and CDF:  
\[f(t)=\lambda\exp(-\lambda t), \tag{12}\]  
\[\omega(t)=\theta\exp(-\theta t), \tag{13}\]  
\[\Omega(t)=1-\exp(-\theta t), \tag{14}\]where \(\lambda\) and \(\theta\) are, respectively, permanent failure and intermittent fault rate. Under the memoryless property of the exponential distribution of the time to permanent failure, the conditional probabilities \(P^{FP}[(\
u-1)\tau,\
u\tau|\eta]\) and \(P^{TP}[(l-1)\tau,l\tau|\eta]\) do not depend on the time to failure. Therefore, as shown in the study [32] (p. 89), the conditional probabilities of false-positive and true-positive can be presented in the following form:  
\[P^{FP}[(\
u-1)\tau,\
u\tau|\eta]=P^{FP}_{flight}\Big{(}1-P^{FP}_{flight} \Big{)}^{\
u-1}, \tag{15}\]  
\[P^{TP}[(l-1)\tau,l\tau|\eta]=\Big{(}1-P^{FP}_{flight}\Big{)}^{l}, \tag{16}\]  
where \(P^{FP}_{flight}\) is the conditional probability of a false-positive event during flight. In-flight, digital avionics systems can be in one of two modes: operation and testing. In operation mode, each avionics system performs its intended functions. In the testing mode, the avionics system does not operate because of performing self-testing. Let us consider example of a Very High-Frequency Omni-Directional Range (VOR) radio navigation system. In the testing mode, a simulated beacon signal with a given azimuth value is generated in an onboard VOR receiver. Then, this signal is fed to the antenna input of the VOR receiver for processing in the same way as the working signal in the operation mode. The decision on the operability of the VOR receiver measuring path is made by comparing the measured value of the tested parameter with the specified one. The test parameter for the VOR receiver is the azimuth measurement error. The time between in-flight test checks is significantly higher than the transient time in corresponding systems, so such checks can be considered independent. Assume there are \(n\) test checks during the flight and permanent failures have an exponential distribution. Besides, the conditional probability of false-positive for a single checking the system by BITE is \(\alpha\). Then, as shown in the study [32] (p. 90), the following formulas calculate the conditional probabilities of false-positive and true-positive:  
\[P^{FP}_{flight}=1-(1-\alpha)^{n-1}, \tag{17}\]  
\[P^{FP}_{flight}=(1-\alpha)^{n}. \tag{18}\]  
Let's simplify the equation for MTBUR by substituting (12)-(16) to (11). \[