Models that can predict the correct logistics actions that need to be taken to ensure the availability of spare parts during maintenance of railway vehicles are of considerable interest. Since the occurrence of defects is a random phenomenon it is not possible to know in advance what spare parts will be required at any point in time. This situation requires the creation of a stochastic model that can incorporate uncertainty. Deterministic and stochastic models of maintenance logistics are based on the assumption of a non-zero stock level of spare parts. Carrying an extensive inventory of parts requires a large financial outlay and it is still often the case that a required spare part is not immediately available and has to be obtained. This paper shows, provided certain conditions are met, that it is possible to create cost-effective maintenance procedures even if the required spare part is not immediately available. Thus, the amount of money locked up in stock can be effectively regulated, thus reducing operating costs.

The goal of every public transport operator is to not only provide high-quality service, but also to minimize investment and operating costs. A significant proportion of the costs are associated with the acquisition, operation and maintenance of reserve vehicles. If the number of vehicles held in reserve is too high, an operator will incur economic losses because vehicles are underused. Conversely, if the number of reserve vehicles is too low, the quality of service will be reduced due to disruptions in the timetable. To determine the number of vehicles to hold in reserve, a coefficient of availability is commonly used. The coefficient of availability is determined by two parameters: reliability and maintainability. Both parameters have the same physical dimensions, the mean time between failures (MTBF) and mean time to repair, and they are expressed in hours. Vehicle wear, which results during the emergence of failures, is not very dependent on the time of operation but is strongly dependent on the distance travelled. It is therefore appropriate to use mean distance between failures (MDBF) instead of MTBF, because it better describes the reliability of vehicles. Using MDBF, however, means that the coefficient of availability is not considered, because MDBF and MTTR have different physical dimensions. This problem is solved by using a random vector that makes it is possible to determine the number of vehicles to hold in reserve based on the distance travelled and maintenance time. This original approach allows the acquisition of better and more authentic data necessary for an operator's decision-making process. Therefore, the number of required reserve vehicles can be much better planned. Ultimately, this positively affects the quality of services and also the investment and operating costs of the operator.

The second edition of standard ISO 26262 (ed. 2018) for functional safety assessment in the automotive industry requires a hardware evaluation using the probabilistic metric for random hardware failures (P MHF). The standard for mentioned purpose highly recommends the fault tree analysis (FTA) utilization but does not give any specific calculation example. Therefore, this article describes computational procedures with derivation and explanation of mathematical formulas for various hardware architectures of electronic systems. Described formulas consider impact of multiple failures and impact of elf-tests, but formulas are relatively simple. This simplicity allows them to be used in the early stages of hardware development when frequent hardware design changes can be expected. Thus, the article with attached case study is intended not only for scientists but also for developers of critical safety-related electronic systems in the automotive industry. K E Y W O R D S car functional safety, fault tree analysis (FTA), hardware assessment, hardware development, ISO 26262, P MHF 1 | INTRODUCTION As in all other areas of industry, the number of electronic systems in car design is constantly growing. This is closely related to the implementation of industry standards in the field of functional safety, based on the principles described in the basic safety standard EN 61508. In the automotive industry, standard ISO 26262:2018 is already established for vehicles up to 3.5 tons. One of the objectives of this standard is to adjust the procedures for Automotive Safety Integrity Level (ASIL) assessment. Generally, standard ISO 26262 assumes for electrical/electronic systems possibility of occurrence of two types of failures, the systematic failures and the random failures, and standard also sets the procedures for failures elimination. However, random failures cannot be completely eliminated, because of their stochastic nature, but it is possible to assess their expected frequency, respectively probability of occurrence, by applying appropriate computational methods. The standard requirement for validation of hardware of assessed system according to ISO 26262 is to perform a quantitative analysis to demonstrate compliance with specific target values depending on the desired ASIL level in two areas. The first area is the diagnostic coverage, thus the ability to detect dangerous system failures robustly and timely. Diagnostic coverage procedures are well described in ISO 26262 and are complemented by a sample example to help developers. Therefore, in the article, this issue will only be mentioned to the extent necessary to understand other texts.

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