# Through-Life Maintenance Cost of Digital Avionics
## 3 Mathematical Modeling of Avionics Maintenance  
Consider the process of operating and maintaining an LRU in the time interval (0, \(T\)). Suppose that the flight duration is \(T\), and in the interval (0, \(T\)) there are \(M+1\) flights, i.e., \(T=(M+1)\tau\). During any flight, the BITE performs LRU testing, which is imperfect to some degree. Due to imperfection of BITE testing, wrong decisions are possible regarding the condition of LRU. A full group of incompatible events when testing the LRU during flight includes the following: true-positive, false-positive, true-negative, and false-negative. Further, it is assumed that in the LRU during the flight a permanent failure or an intermittent fault may occur. In the case of a permanent failure upon the flight, the onboard computer will disable the LRU or ignore its information. In the case of an intermittent fault in-flight, the LRU usually does not shut down during the flight. Still, the onboard computer will record information about the intermittent fault that occurred during the last flight. Thus, after aircraft landing, the LRU is demounted from the aircraft board on one of the following occasions:  
* the event of a \"false-positive\" occurred in the last flight,
* the event of a \"true-negative\" happened during the last flight,
* an intermittent fault occurred in the previous flight. The expected maintenance cost (EMC) is dependent on the chosen maintenance option for a specific period of the lifetime. The EMC, in general, we calculate as follows:  
\[E(MC_{i})=q\times N\times EC_{r,i}\times ER(T)+CAPEX_{i}, \tag{1}\]  
where \(q\) is the quantity of identical LRU on the aircraft board, \(N\) is the quantity of aircraft in the fleet of the airline, \(EC_{r,i}\) is the expected cost of repairing the LRU removed from the aircraft board for the maintenance option number \(i\), \(ER(T)\) is the expected number of the unscheduled LRU removals during time \(T\), \(CAPEX_{i}\) is the capital expenditures for the maintenance option number \(i\). As shown in (1), parameters \(q\), \(N\), and \(ER(T)\) do not depend on the selected maintenance option. Parameters \(EC_{r,i}\) and \(CAPEX_{i}\) rely on the maintenance organization and capital expenditures introduced to the diagnostic equipment and spare parts. First of all, let us determine the \(ER(T)\). The expected number of the unscheduled LRU removals in the time span (0, \(T\)) is the ratio of \(T\) to the MTBUR, i.e.,  
\[ER(T)=T/E(TBUR_{T})\,, \tag{2}\]  
where \(E(TBUR_{T})\) is the MTBUR on a finite time horison (0, \(T\)). MTBUR is one of the essential indicators of avionics operational reliability. This indicator depends on the characteristics of the LRU permanent failure-free and intermittent fault-free operation and the validity of the in-flight BITE functioning. Under [28], the MTBUR is about 50% of the MTBF for modern avionics, where MTBF is the mean time between permanent failures. This difference between MTBUR and MTBF leads to increased direct operating costs of about 40% [29]. Let's determine an equation for MTBUR considering the main causes of LRU dismounting from the aircraft board, including failures, faults, and the false-positives of BITE during the flight. Let the permanent failure arises at the time \(\text{H}=\eta\), where \(k\tau<\eta\leq(k+1)\tau\), \(k=\overline{0,M}\) and H is the random time to permanent failure. Consider the events leading to the unscheduled LRU removal from the aircraft board. If a false-positive occurs during the \(\
u\)-th flight the LRU will operate till the time \(\
u\tau\), where \(\
u=\overline{1,k}\).