# Semi-Markov Model of the System of Repairs and Preventive Replacements by Age of City Buses
## 1 Introduction  
The basic task of transport systems is to transport people, animals and goods. The transport tasks, due to their particular specification, are carried out by different types of transport means. A very important branch of the transport system is the road passenger transport, which can generally be divided into international transport, interurban transport and urban transport. Urban transport systems usually operate in medium and large cities, in suburban areas and in industrialised areas. One of the important types of urban transport is the urban bus transport. The task of this type of transport system is to reliably and punctually carry out transport tasks along defined routes in accordance with an accepted timetable of courses [1] The basic characteristics for evaluating the functioning of this type of transport system are economic efficiency characteristics (e.g., profit per unit time) and operational-technical efficiency characteristics (e.g., readiness of the city buses to carry out the assigned transport tasks) [2, 3]. Any kind of disruption in the implementation of the assigned transport tasks, including downtime caused by damages to the means of transport (city buses) causes a decrease in the reliability and readiness of technical facilities and generates additional costs (losses). These losses arise as a result of corrective maintenance (CM) conducted after damage, losses caused by fines for non-performance of transport services, and costs related to the maintenance of reserve buses whose task is to replace damaged buses. One of the ways of ensuring the correct and efficient fulfilment of transport tasks in urban bus operation systems is the implementation of preventive maintenance (PM). The implementation of these activities consists of planning the timing and scope of preventive maintenance in such a way as to keep their costs lower than the costs of repairs after the damage. For this reason, the determination of optimal times forpreventive maintenance of technical objects is an important problem of planning strategies in the systems of exploitation of the means of transport [4]. During the operation of technical objects, their elements are subject to wear and tear processes and the impact of external factors, which causes damage to these objects. The resulting damages are the cause of lowering the effectiveness of functioning of the analysed systems. In order to ensure an appropriate level of reliability of technical facilities, different types of strategies are applied in the subsystems for ensuring serviceability. These activities are divided into two types: preventive maintenance and corrective maintenance. In practice, corrective maintenance is carried out in two variants. Firstly, the perfect repair (PR), which makes the system \"As-Good-As-New\" (AGAN), and secondly the minimal repair (MR), which makes the system \"As-Bad-As-Old\" (ABAO) are conducted. Normally, CM corrective repair costs and times are higher than PM preventive repair costs. This is due to the fact that, in general, CM activities require prior identification of the damage and high skills of personnel (diagnostics and mechanics). In addition, there are the costs associated with the unplanned downtime of technical facilities caused by the damage to them. From this, it follows that it is possible to plan preventive actions (the scope and frequency of preventive repairs and replacements) in such a way as to ensure the required level of readiness of technical facilities and to reduce system maintenance costs. This requires the development and application of rational preventive repair and replacement strategies. For these reasons, the development of various preventive action strategies with the application of optimal decision models to reduce the system maintenance costs and the risk of adverse events is an important research topic in reliability engineering. For the first time, the concept of minimal repair can be found in [PERSON]'s study [5].