# Through-Life Maintenance Cost of Digital Avionics
## 3 Mathematical Modeling of Avionics Maintenance
By the formula of mathematical expectation for a discrete random variable, we determine that  
\[E\big{(}TBUR_{T}^{FP}|\eta,\,\eta\leq T\,\,\big{)}=\sum_{\
u=1}^{k}\
u\tau \text{P}^{FP}[(\
u-1)\tau,\
u\tau|\eta][1-\Omega(\
u\tau)], \tag{3}\]  
where \(E(TBUR_{T}^{FP}|\eta)\) is the conditional mathematical expectation of the time to unscheduled removal due to false-positive, \(\,\text{P}^{FP}[(\
u-1)\tau,\
u\tau|\eta]\,\) is the conditional probability of false-positive during \(\
u\)-th flight provided that H = \(\eta\), and \(\Omega(t)\) is the cumulative distribution function (CDF) of the time to intermittent fault. Besides, the LRU will be removed at time \(l\tau\) if during the \(l\)-th flight an intermittent fault occurs, where \(l=\overline{1,k}\). By the formula of mathematical expectation for a discrete random variable, we obtain  
\[E\big{(}TBUR_{T}^{IF}|\eta,\eta\leq T\,\big{)}=\sum_{l=1}^{k}l\tau\text{P}^{ TP}[(l-1)\tau,l\tau|\eta]\int\limits_{(l-1)\tau}^{l\tau}\omega(x)dx, \tag{4}\]  
where \(E(TBUR_{T}^{IF}|\eta)\) is the conditional mathematical expectation of the time to unscheduled removal due to intermittent fault, \(\,\text{P}^{TP}[(l-1)\tau,l\tau|\eta]\,\) is the conditional probability of true-positive during \(l\)-th flight provided that H = \(\eta\), and \(\omega(t)\) is the probability density function (PDF) of the time to intermittent fault. Finally, if there was no BITE error like false-positive or intermittent fault before the moment \(\eta\), then the LRU will be dismantled after the aircraft landing at the moment \((k+1)\tau\). In this case, the expected time till LRU removal is given by  
\[E\big{(}TBUR_{T}^{PF}|\eta,\,\eta\leq T\,\big{)}=(k+1)\tau\text{P}^{TP}[(k-1) \tau,k\tau|\eta][1-\Omega(k\tau)]. \tag{5}\]  
Now assume that \(\eta>T\). In this case, the following formulas determine the conditional mathematical expectation of the time to unscheduled removal:  
- due to false-positive  
\[E\big{(}TBUR_{T}^{FP}|\eta,\,\eta>T\,\,\big{)}=\sum_{k=1}^{M}\
u\tau\text{P}^{ FP}[(k-1)\tau,k\tau|\eta][1-\Omega(k\tau)], \tag{6}\]  
- due to intermittent fault  
\[E\big{(}TBUR_{T}^{IF}|\eta,\eta>T\,\big{)}=\sum_{k=1}^{M}k\tau\text{P}^{TP}[(k -1)\tau,k\tau|\eta]\int\limits_{(k-1)\tau}^{k\tau}\omega(x)dx, \tag{7}\]  
- due to permanent failure:  
\[E\big{(}TBUR_{T}^{PF}|\eta,\,\eta>T\,\big{)}=T\times\text{P}^{TP}[(M-1)\tau,M \tau|\eta][1-\Omega(T)]. \tag{8}\]  
Thus, assuming that H = \(\eta\) and considering (3)-(8), the expected time between unscheduled LRU removals is \[E(TBUIR_{T}|\eta)=\left\{ \begin{array}{c}\sum\limits_{\
u=1}^{k}\
u\tau P^{FP}[( \
u-1)\tau,\
u\tau|\eta][1-\Omega(\
u\tau)]+\sum\limits_{l=1}^{k}l\tau P^{TP}[( l-1)\tau,l\tau|\eta]\int\limits_{(l-1)\tau}^{l\tau}\omega(x)dx+\\ \\ (k+1)\tau P^{TP}[(k-1)\tau,k\tau|\eta][1-\Omega(k\tau)],\\ \\ \text{if}\,k\tau<\eta\leq(k+1)\tau,k=\overline{0,M},\\ \\ \sum\limits_{k=1}^{M}\
u\tau P^{FP}[(k-1)\tau,k\tau|\eta][1-\Omega(k\tau)]+ \sum\limits_{k=1}^{M}k\tau P^{TP}[(k-1)\tau,k\tau|\eta]\int\limits_{(k-1)\tau}^ {k\tau}\omega(x)dx+\\ \\ T\times P^{TP}[(M-1)\tau,M\tau|\eta][1-\Omega(T)],\,\text{if}\,\eta>\,T\end{array} \right. \tag{9}\]  
We define the conditional probability \(P^{FP}[(\
u-1)\tau,\
u\tau|\eta]\) as the probability of mutual occurrence of the following events: when tested during \(\
u-1\) flights, BITE judged the LRU as operable, and during the \(\
u\)-th flight BITE recognized the LRU as inoperable, provided that H = \(\eta\). Analogically, the conditional probability \(P^{TP}[(l-1)\tau,l\tau|\eta]\) presents the probability of mutual occurrence of the following events: when tested during \(l\) flights, BITE judged the LRU as operable, provided that H = \(\eta\). We determine MTBUR by the formula of total expectation of continuous random variable, which is random time to permanent failure  
\[E(TBUIR_{T})=\sum\limits_{k=0}^{M}\int\limits_{k\tau}^{(k+1)\tau}E(TBUIR_{T}| \eta)f(\eta)d\eta, \tag{10}\]  
where \(f(\eta)\) is the PDF of the time to permanent failure.