## 1 Introduction  
By the end of 2020, the total mileage of urban rail transit operations in mainland China reached 7969.7 km. Among them, the mileage of subway operating lines reached 6280.8 km. The amount of passenger traffic in 2020 reached 17.59 billion, which means that more than 24 million people travel by subway every day [1]. Drive safety is one of the inevitable issues with the surge in subway operations. The braking system is one of the most important subsystems related to the driving safety of the subway, and untimely braking is the biggest safety hazard during subway operation [2, 3]. Subway braking activities are divided into general braking and emergency braking [4]. General braking is also known as braking when the subway stops at a station, while emergency braking is braking under abnormal conditions while the subway is in motion. Nowadays, the primary source of braking force for general braking and emergency braking is the electric braking system [5]. The electric brakes are activated after the train control system issues a braking command, then the subway vehicle is ensured to be slowed down immediately. The electric brake system braking force is provided by the torque generated by the brake motor and the corresponding mechanical mechanism, the braking output can be adjusted, and the emergency braking conditions will require the brake system to provide the maximum output. The integrity of the emergency brake function affects the safety of the train, while the general brake does not. The braking activity explored in this paper refers exclusively to emergency braking. The brake motor as a part of the components forms the braking system and governs the key functions of the braking system [5]. Brake motors are generally installed in the bottom of multiple carriages in a distributed manner, with multiple motors working together to provide the braking force needed to brake, as is shown in Figure 1. If one or more of the motors fail to work, the maximum braking force provided by the braking system will be reduced. If the maximum braking force drops to a certain level, the emergency braking distance of the train will not meet a safety threshold, which is called the safe braking distance. Therefore, to meet this safe braking distance requirement, a sufficient number of brake motors are required to work properly. Additionally, the greater the residual redundance before the system fails, the more effective the brakes will be. Such a braking system is a typical K-out-of-N system. The K-out-of-N system is a common type of redundant system where the failure of components leads to system degradation, wherein the system as a whole fails when K components fail. To keep the train brakes at a safe distance during operation, the brake system needs maintenance to repair the failed motors. The maintenance of the train system generally includes daily maintenance and overhaul. Daily maintenance is small-scale maintenance, mainly following the list of key components to inspect and simple repair, while overhaul is a complete inspection and replacement of the system. Different maintenance strategies can recover the degenerated braking distance to different extents. Thus, the system can operate properly while meeting safety conditions. Under the combination of routine maintenance and overhaul strategy, the better the effect of minor repair, the better the overhaul cycle can be properly extended, and vice versa. The coordination of routine maintenance and overhaul cycles is a solution to the balance between operational input and operational safety. Therefore, considering this mixed maintenance characteristic of train braking systems to ensure safety results in more economical operation for the subway corporations. It is necessary to develop an analytical model of the K-out-of-N system comprehensively including both cycles to address the balance between operation and safety, which is completely a new issue. For train braking system performance, existing models have been studied based on physical models [6,7].