## 2 Problem Description  
_System Description_  
The metro train braking system in this paper is modeled on its brake motors. Within the braking system, the total number of brake motors is n and the brake motors are all intact at the beginning. All brake motors have the same failure rate. The braking system needs to satisfy the condition of safe braking distance, that is, to ensure that k brake motors can work properly. Such a system is a typical K-out-of-N: G system.  
For the metro train braking system, it is given that its work and preventive daily maintenance are intermittent and alternating in cycles of days. The operation hours of the train in a day are fixed \(\tau_{1}\), and daily maintenance cannot be performed during operation, so the redundant degree of the brake motors will only gradually decrease due to failure. The non-working hours of the train are considered daily maintenance time \(\tau_{2}\). During the daily maintenance period, the brake motors are stopped or on standby for inspection and no failure occurs, so the redundancy of the brake motors will rise with the maintenance.  
In addition to the daily maintenance, the system schedules overhaulst. The overhaul interval is T, and T must be an integer multiple of \(\tau=\tau_{1}+\tau_{2}\). After the overhaul, the system will be restored to a defined state between state n and state \(k+1\).  
While a brake motor in the train braking system fails, the system is still available if the remaining redundancy of the brake motors is not completely lost, i.e., if there is still a remaining available redundancy. If the failed brake motors are not fully repaired, the system is in an incomplete state of operation. Thus, the train braking system contains \(n-k+1\) states. The state of this braking system at a time \(t\) is described by the number of intact brake motors \(n(t)\) at that moment.  
The K-out-of-N system can operate when \(k\) of the \(n\) components in the system are working properly. All of the components are intact in the starting phase.  
Thus, the following assumptions are made in this paper:  
1. The failure time of each brake motor follows an exponential distribution, which is a common assumption in K-out-of-N systems. All component failures are independent.
2. All brake motors of the system are in operating condition at the outset. Additionally, when a component fails, it will immediately lose its ability to work.
3. The work and maintenance of the system are carried out at intervals and alternates on a fixed cycle. This is in consideration of the characteristics of the maintenance schedule possessed by such equipment as trains.
4. During system operation, inspection and maintenance activities for motor failures cannot be performed. Inspection and repair of motor failures can be performed only during maintenance time. The maintenance time of the brake motors also follows an exponential distribution and is independent.
5. It is assumed that the system has sufficiently high safety requirements in the design phase to guarantee that the system will not experience system failures during operation. It means that the braking system will not fail during operation.
6. During the overhaul, the components are not repaired one by one. Therefore, the overhaul time and the probability of change in the system's state during the overhaul are not considered. Only the system's repair effect after the overhaul will be considered.  
After finding the braking system state probability, we then collect the train operation data to calculate the train braking distance, coupling the braking system state probability and train braking distance to obtain the braking distance with the brake motor state change relationship and braking distance expectation.