# Semi-Markov Model of the System of Repairs and Preventive Replacements by Age of City Buses
## 1 Introduction
Other papers consider policies that assume randomness of model parameters, e.g., papers [23; 24] assume random repair costs. On the other hand, the paper [25] describes an age replacement policy with Bayesian imperfect repair model, in which the probability of an exact repair is a random variable with a specified distribution. A similar approach is adopted in the paper [26], where the optimal age replacement policy is determined in the case of minimising the cost per unit time. The results were obtained both for an infinite time horizon and for a single replacement cycle. The sequential imperfect preventive maintenance model for city buses is presented in the paper [27]. In this model, the optimal decision-making concerning the efficiency of maintenance of city buses is realized on the basis of the evaluation of the difference between the actual and expected increments of the intensity of damage. The results presented in this paper are a continuation of the considerations presented in papers [4; 28; 29]. Similarly, to this paper, the results presented in these papers were obtained based on the study of the semi-Markov models. In the paper [4] a 4-state model of replacements according to the age of technical objects with a guarantee was analysed, in which the criterion function is the cost of preventive replacement determined per unit of time, while in the paper [28] a multi-state model of exploitation decisions was developed, in which corrective repairs (after a damage) and preventive replacements are carried out, and the profit per unit of time was used as the criterion function. A direct continuation of the conducted research are the results presented in the paper [29], in which a 4-state model of a service system with minimum repair was considered. The model was developed with the application of the theory of semi-Markov processes, and the theoretical considerations were illustrated with numerical examples on the basis of the assumed sample data. In this paper, on the other hand, the 5-state semi-Markov model of the system of preventive repairs and replacements according to the age of city buses is considered. The aim of the paper is to develop and study a semi-Markov model of preventive repairs and replacements according to the age of city buses using two criterion functions: profit per unit time and the coefficient of readiness of city buses to carry out the assigned transport tasks, and also to formulate sufficient conditions for the existence of the maximum of these functions. The results of studies of the model, developed on the basis of real data, can be the basis for decision-making in the analysed system of exploitation of urban transport means. In the developed mathematical model, the basis for the construction of the criterion function is the limit theorem for semi-Markov processes [30; 31]. In the developed model, four types of implemented urban bus renewal processes are considered. The conditions for the existence of the extremum (maximum) of the criterion functions were formulated for the assumed assumptions. The theoretical considerations presented in the paper are illustrated by the results of calculations. The calculation examples have been developed on the basis of operational data obtained from a real urban bus operation system. In the first example, for the estimated input data, the profit per unit time and the value of the readiness factor are maximised. In the second example, the analysed criterion functions are investigated, in case when the number (frequency) of preventive replacements will be higher than in the first example. In both the calculation examples, it is assumed that the time to failure of the technical object (city bus) has a Weibull distribution.