# Through-Life Maintenance Cost of Digital Avionics
## 4 Warranty Maintenance Models of Avionics Systems
We determine \(EC_{W,1}\) by the following formula:  
\[EC_{W,1}=q\times LC\times I_{M}^{O-level}ER(T_{W})+(PS+US)\times C_{LRU}/N_{W}, \tag{20}\]  
where \(q\) is the quantity of the same LRU in the onboard electronic system, \(LC\) is the operational maintenance labor cost per hour ($/h), \(t_{M}^{O-level}\) is the mean time of O-level maintenance, \(ER(T_{W})\) is the average number of unscheduled removals of LRU due to permanent failures, intermittent faults, and in-flight false-positives for time \(T_{W}\), \(N_{W}\) is the quantity of airplanes under the supplier's warranty, \(PS\) is the planned quantity of spare LRU in the warehouse, \(US\) is the unplanned quantity of spare LRU that will need to be supplied from the manufacturer to provide flight regularity, and \(C_{LRU}\) is the cost of a spare LRU. We assume that the airline has ground test equipment at the I-level maintenance for rechecking the LRU demounted from the aircraft board in the second WM option. Since conventional test equipment cannot practically detect the presence of intermittent faults in dismounted LRU, such units are sent to the warehouse after testing. We denote the average cost associated with the second WM option as \(EC_{W,2}\). It includes the cost components due to dismounting and installing LRU on aircraft board, rechecking dismounted LRU by ground test equipment, ground test equipment cost, and cost of spare LRU in the warehouse recalculated per aircraft. The following formula determines \(EC_{W,2}\):  
\[EC_{W,2}=q\times LC\left(t_{M}^{O-level}+t_{TE}^{I-level}\right)ER(T_{W})+\frac{C_ {TE}^{I-level}}{N_{W}\times F_{TE}^{I-level}}+(PS+US)C_{LRU}/N_{W}\,, \tag{21}\]  
where \(t_{TE}^{I-level}\) is the average LRU testing time at the maintenance on the I-level, \(F_{TE}^{I-level}\) is the quantity of different LRU types that the ground test equipment can test at the maintenance on the I-level, \(C_{TE}^{I-level}\) is the ground test equipment cost on the I-level. We can see from (20) and (21) that \(EC_{W,1}\) and \(EC_{W,2}\) depend on the quantities of spare LRU \(PS\) and \(US\). Further, we will compute the optimal quantity of spare LRU by using the following criterion [24]:  
\[\left[t_{spare}(PS,US)+t_{M}^{O-level}-t_{stop}\right]\to 0, \tag{22}\]  
where \(t_{spare}\) is the average waiting time for a spare LRU from the warehouse in the situation \"aircraft on the ground\", \(t_{stop}\) is the average time of the scheduled stop of the aircraft at the airport of the airline's base. We can see from (22) that if \(t_{stop}\geq t_{spare}(PS,US)+t_{M}^{O-level}\), then no disruption of the regularity of flights will occur. In contrast, if \(t_{stop}<\Delta t_{spare}(PS,US)+t_{M}^{O-level}\), there will be a disruption of the flight regularity. Therefore, the optimal quantity of spare LRU is the minimum possible number of units, which ensures that there are no delays in departures from the airline base airport. We used the continuous-time Markov chain to describe the operation of the warehouse of spare LRU at the airline base [25]. Let's evaluate the mean time of LRU repair \(E(T_{repair})\). For the first WM option, we have  
\[E\big{(}T_{repair}\big{)}=T_{RS}. \tag{23}\]  
For the second WM option, only those LRUs are shipped to the manufacturer for repair, the permanent failure of which is confirmed by checking using ground test equipment. Therefore, the time \(E(T_{repair})\) will be less than for the first WM option. We compute the time \(E(T_{repair})\) by the formula for a discrete random variable's mathematical expectation.