# Semi-Markov Model of the System of Repairs and Preventive Replacements by Age of City Buses
## 1 Introduction
In this study, a repair model is considered in which the criterion function is the monthly revenue generated for the technical facility under consideration. However, this model was developed on the basis of the queueing theory and not the reliability theory. However, the concept of minimum repairs in relation to the reliability theory was introduced by [PERSON] and [PERSON] in their paper [6]. In this paper, the model of periodic replacements and minimum repairs is considered, in which it is assumed that after each minimum repair the damaged technical system is restored only to the same failure condition as before the damage. In a formal way, the concept of minimum repairs was defined by [PERSON] and [PERSON] in their paper [7]. In the paper [8] [PERSON] and [PERSON] also consider the issue of minimal repair. In this paper, the authors assume that when a technical object is damaged, a perfect repair is performed with probability \(\mathrm{q}=1-\mathrm{p}\). A modified version of such a model was proposed by [PERSON] and [PERSON] [9]. In the model developed, the object is replaced with a new one after time \(\mathrm{T}\) and either a perfect repair or a minimal repair is performed with probabilities \(\mathrm{p}\) and \(\mathrm{q}\), respectively, for intermittent failures. In the literature one can find descriptions of models of systems with minimum repairs, which have been developed with the use of various methods and mathematical tools. An overview of the used modelling methods and the construction of criterion functions in models of minimum repairs with preventive maintenance can be found, for example, in the papers [10; 11]. The papers classify and discuss models of maintenance strategies for technical objects developed for both finite and infinite time horizons, in which the criterion functions are total costs, unit costs, reliability and readiness. Most of the models presented in the literature have been developed on the basis of renewal theory, while less frequently with the use of stochastic processes, including Markov and semi-Markov process models. For example, in the paper [12] the criterion functions cost per unit time and system availability were determined on the basis of a semi-Markov model in an infinite time horizon, and in the paper [13] the model of the imperfect maintenance system was developed using the theory of Markov processes, and the readiness function is a criterion for optimisation. In practice, the effectiveness of the realised repair is between AGAN and ABAO repair and it concerns the so-called imperfect maintenance/repairs. The methods concerning preventive repairs and replacements using the repair mechanism with an imperfect maintenance model with the (p, q) rule are extensively discussed in the paper [10]. The paper [14] presents the problem of imperfect repair with periodic preventive replacement. Models of preventive replacements by age are presented in papers [15; 16]. In this type of model, it was assumed that the probabilities p and q depend on the age of the technical object at the time of failure, and that a thorough repair restores the technical object to the reliability state as for a new object, while a minimal repair restores the technical object to the reliability state just before failure. In the paper [17] it was shown that the PM policy limiting the possibility of failure can be more cost-effective than the PM policy implemented according to age, while the authors of the papers [18; 19] analysed the imperfect repair system model with a delayed time concept. Models of imperfect maintenance systems, using different age replacement policies that take into account different types of repairs after failure and their cost structures, have been presented in a number of papers. In [20], the authors consider replacement policies depending on the age of the system and the minimisation of repair costs. The authors of the papers [21; 22] consider the age replacement policy of system subject to shocks in their models.