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1

Semi-Markov approach for reliability modelling of light utility vehicles

Oszczypała Mateusz, Ziółkowski Jarosław, Małachowski Jerzy

Eksploatacja i Niezawodność

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2023

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Vol. 25, no. 2

art. no. 161859

EN

Vehicles are important elements of military transport systems. Semi-Markov processes, owing to the generic assumption form, are a useful tool for modelling the operation process of numerous technical objects and systems. The suggested approach is an extension of existing stochastic methods employed for a wide spectrum of technical objects; however, research on light utility vehicles complements the subject gap in the scientific literature. This research paper discusses the 3-state semi-Markov model implemented for the purposes of developing reliability analyses. Based on an empirical course of the operation process, the model was validated in terms of determining the conditional probabilities of interstate transitions for an embedded Markov chain, as well as parameters of time distribution functions. The Laplace transform was used to determine the reliability function, the failure probability density function, the failure intensity, and the expected time to failure. The readiness index values were calculated on ergodic probabilities.

2

Evaluation of the maintenance system readiness using the semi-Markov model taking into account hidden factors

Kozłowski Edward, Borucka Anna, Oleszczuk Piotr, Jałowiec Tomasz

Eksploatacja i Niezawodność

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2023

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Vol. 25, no. 4

art. no. 172857

EN

Modelling the time that the system remains in a given state using classical distributions is not always possible. In many cases, empirical distributions are multimodal due to the influence of external, hidden factors and the selection of the best classical distributions may lead to erroneous results. In the article the method of diagnosis of influence of hidden factors into sojourn time of semi-Markov models was presented. In order to capture hidden factors, the authors proposed to model the distributions of the sojourn time with a mixture of distributions, which is a significant novelty in relation to the studies presented in the literature. Hidden factors directly affect the reliability of technical systems. Detecting the existence of these factors enables more accurate modeling of system readiness. Paying attention to irregularities caused by hidden factors makes it possible to reduce system maintenance costs. Such a system model providescomplete information and enables a reliable assessment of the system readiness and maintenance.

3

The matter of decision-making control over operation processes of marine power plant systems with the use of their models in the form of semi-Markov decision-making processes

Girtler Jerzy, Rudnicki Jacek

Polish Maritime Research

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2021

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nr 1

116--126

EN

The article presents the possibility to control the real operation process of an arbitrary device installed in the marine power plant based on the four-state semi-Markov process, being the model of the process, which describes the transition process of operational states of the device (ek , k = 1, 2, 3, 4), and the transition process of its technical states (sl , l = 1, 2, 3). The operational states ek (k = 1, 2, 3, 4) have the following interpretation: e1 – active operation state resulting from the task performed by the device, e2 – state of ready-to-operate stop of the device, e3 – state of planned preventive service of the device, e4 – state of unplanned service of the device, forced by its damage. Whereas the interpretation of the technical states sl (l = 1, 2, 3) is as follows: s1 – state of full serviceability of the device, s2 – state of partial serviceability of the device, and s3 – state of unserviceability of the device. All these states are precisely defined for the ship main engine (SG). A hypothesis is proposed which justifies the use of this model to examine real state transitions in marine power plant device operation processes. The article shows the possibility to make operating decisions ensuring a rational course of the device operation process when the proposed model of this process and the dynamic programming method based on the Bellman’s principle of optimality are applied. The optimisation criterion adopted when making operating decisions is the expected profit to be gained as a result of functioning of the device in the time interval [τ0 , τm], being the sum of the expected profit gained in interval [τ0 , τ1 ] and to be gained in interval [τ1 , τm].

4

Modelling of transport system operational reliability a Markov approach

Smolarek Leszek, Ziemska Monika

Journal of KONES

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2019

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Vol. 26, No. 4

227--234

EN

The workability of a transport system is associated with performance and operational reliability. Operational reliability provides a measure of the probability that a transport system will realize transport process as intended. Performance reliability is an adequacy measure of transport process realization under specific environmental and traffic conditions. Transport system can be modelled as repairable, multistate, non-homogenous rectangular or dendrite system. This article provides the Markov and semi Markov models for estimation of the operational and performance reliability of city transport system. The system is semi homogenous it means that serial subsystems have the same reliability function. The reliability of any serial subsystem is exponential. The distribution of the repair time is any probability distribution. In case where the probability distribution of the repair time is exponential, the Markov process is used to construct simulation model. The simulation model was applied at Microsoft Excel. Many stochastic models in engineering, logistic and even finance or insurance are setup in a spreadsheet for simulation. The semi Markov model of the multistate reliability of repaired system is applied to the street system. The embedded Markov chain was used to count stationary probabilities. The possibility of application of the results is illustrated by an example for the systems with rectangular or dendrite shaped accordingly, consist of three types of elements.

5

Forecasting the readiness of special vehicles using the semi-Markov model

Borucka Anna, Niewczas Andrzej, Hasilova Kamila

Eksploatacja i Niezawodność

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2019

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Vol. 21, no. 4

662--669

EN

The vehicle exploitation system, consisting of statistically identical objects that perform intervention tasks, not subject to systematic changes, can be modelled as a stationary stochastic process. Such a model allows to determine the probabilistic indicators of current and boundary readiness of the system. This article presents the use of the semi-Markov process, based on three operating states: operation, ready-to-be-used and repair, to study a transport system consisting of special vehicles. On the example of a sample consisting of police patrol cars, experimental studies of the intensity of fleet utilization, time of failure-free operation of vehicles were carried out, and it was demonstrated that the examined transport system is characterized by a satisfactory, stationary readiness coefficient. The developmental possibilities of the presented modelling method were emphasized.

PL

System eksploatacji samochodów, które realizują zadania interwencyjne, niepodlegający systematycznym zmianom może być modelowany jako stacjonarny proces stochastyczny. Taki model pozwala wyznaczyć probabilistyczne wskaźniki bieżącej i granicznej gotowości systemu. W niniejszym artykule, do modelowania systemu eksploatacji pojazdów specjalnych, wykorzystano proces semi-Markowa, oparty na trzech stanach eksploatacyjnych: użytkowania, postoju użytkowego i naprawy. Na przykładzie próby radiowozów policyjnych przeprowadzono doświadczalne badania intensywności użytkowania floty, czasu bezawaryjnej pracy pojazdów a także wykazano, że badany system transportowy charakteryzuje się zadowalającym, stacjonarnym współczynnikiem gotowości. Podkreślono rozwojowe możliwości przedstawionej metody modelowania.

6

Semi-Markov Reliability Model of System Composed of Main Subsystem, Cold Backup Component and Switch

Grabski F.

Journal of Polish Safety and Reliability Association

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2017

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Vol. 8, No. 1

47--54

EN

Probabilistic model of a system composed of a main component, an emergency backup component and the automatic switch are discussed in this paper. The reliability model is semi-Markov process describing evolution of the system. Conditional time to failure of the system is represented by a random variable denoting the first passage time of the process from the given state to the subset of states. The appropriate theorems of the Semi-Markov processes theory allow us to evaluate the reliability function and some reliability characteristics. To calculate the reliability function and mean time to failure of the system we apply theorems of the Semi-Markov processes theory concerning the conditional reliability functions.

7

Semi-Markov model of multi-modal transport operation

Grabski F.

Journal of KONES

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2017

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Vol. 24, No. 4

47--54

EN

Multi-modal transport means the transport of the objects through at least two different carriers of any combination of simple tasks of transport carriers (by truck, by train, by ship or by plane). A Semi-Markov (SM) model of multi-modal transport operation is presented in the article. The SM process is defined by the renewal kernel of that one. In our model, time to failure of the operation is represented by a random variable that denotes the first passage time from the given state to the subset of states. The duration of one operation cycle is a random variable representing the return time to the initial state. The appropriate theorems of the Semi-Markov processes theory allow us to calculate characteristics and parameters of the transport operation model. The article presents the example of the transport operation final part of container with cargo from Warsaw to Stockholm, where from Warsaw to Gdynia, the container is transported by lorry, from Gdynia to Karlscorona by ferry and from Karlscorona to Stockholm by truck.

8

The renewal process generated by return times of semi-Markov process in reliability models

Grabski F.

Journal of KONBiN

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2017

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No. 43

365--380

EN

The renewal process generated by the return times of semi-Markov process to a given state is considered in the paper. The return time to a state j and also a first passage time from a given state i to the state j of semi-Markov process are basic concepts that are used to determine this process. The systems of equations for distributions, expectations and second moments of these random variables are presented. Theorem concerning the asymptotic distribution of the considered renewal process is presented in this article. Moreover an illustrative example from the reliability theory is presented in the paper.

PL

W artykule jest rozważany proces odnowy generowany przez czasy powrotu procesu semi-Markowa (SM) do danego stanu. Czas powrotu do stanu j, a także czas pierwszego przejście od danego stanu i do stanu j semi-Markowa procesu są podstawowymi pojęciami, które są stosowane do określenia tego procesu. W pracy zostały przedstawione układy równań dla transformat rozkładów, wartości oczekiwanych i drugich momentów tych zmiennych losowych. Twierdzenie dotyczące asymptotycznego rozkładu badanego procesu odnowy jest kluczowym elementem pracy. Ponadto został przestawiony przykład z teorii niezawodności ilustrujący rozważane problemy.

9

Study of the seismic activity in central Ionian Islands via semi-Markov modelling

Pertsinidou Ch.E., Tsaklidis G., Papadimitriou E.

Acta Geophysica

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2017

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Vol. 65, no. 3

533--541

EN

The seismicity of the central Ionian Islands (M ≥ 5.2, 1911–2014) is studied via a semi-Markov chain which is investigated in terms of the destination probabilities (occurrence probabilities). The interevent times are considered to follow geometric (in which case the semi-Markov model reduces to a Markov model) or Pareto distributions. The study of the destination probabilities is useful for forecasting purposes because they can provide the more probable earthquake magnitude and occurrence time. Using the first half of the data sample for the estimation procedure and the other half for forecasting purposes it is found that the time windows obtained by the destination probabilities include 72.9% of the observed earthquake occurrence times (for all magnitudes) and 71.4% for the larger (M ≥ 6.0) ones.

10

Reliability and maintainability characteristics in semi-Markov models

Grabski F.

Journal of Polish Safety and Reliability Association

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2016

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Vol. 7, No. 1

79--86

EN

The characteristics of semi-Markov process we can translate on the reliability characteristics in the semi-Markov reliability model. The cumulative distribution functions of the first passage time from the given states to subset of states, expected values and second moments corresponding to them which are considered in this paper allow to define reliability function of the system. The equations for many reliability characteristics and parameters are here presented.

11

Prediction of climate-weather change process for port oil piping transportation system and maritime ferry operating at Baltic Sea area

Kołowrocki K., Soszyńska-Budny J.

Journal of Polish Safety and Reliability Association

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2016

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Vol. 7, No. 3

143--148

EN

There are presented the methods of prediction of the climate-weather change process. These are the methods and procedures for estimating the unknown basic parameters of the climate-weather change process semi-Markov model and identifying the distributions of the climate-weather change process conditional sojourn times at the climate-weather states. There are given the formulae estimating the probabilities of the climate-weather change process staying at the particular climate-weather states at the initial moment, the probabilities of the climate-weather change transitions between the climate-weather states and the parameters of the distributions suitable and typical for the description of the climate-weather change process conditional sojourn times at the particular climate-weather states. The proposed statistical methods applications for the unknown parameters identification of the climate-weather change process model determining the climate-weather change process parameters for the initial point of the port oil piping transportation system are presented.

12

Modelling climate-weather change process including extreme weather hazards for maritime ferry

Jakusik E., Kołowrocki K., Kuligowska E., Soszyńska-Budny J., Torbicki M.

Journal of Polish Safety and Reliability Association

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2016

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Vol. 7, No. 3

41--46

EN

The climate-weather change process for the maritime ferry operating at Port Gdynia and at Baltic Sea open waters between Gdynia bay and Karlskrona bay is considered and its states are defined. Further, the semi-Markov process is defined and used to create a general probabilistic model of the climate-weather change process for the maritime ferry operating at considered areas.

13

Modelling climate-weather change process including extreme weather hazards for port oil piping transportation system

Jakusik E., Kołowrocki K., Kuligowska E., Soszyńska-Budny J., Torbicki M.

Journal of Polish Safety and Reliability Association

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2016

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Vol. 7, No. 3

31--40

EN

The climate-weather change process for the critical infrastructure operating area is considered and its states are defined. Further, the semi-Markov process is used to create a general probabilistic model of the climate-weather change process for the critical infrastructure operating area. To construct this model the vector of probabilities of the climate-weather change process staying at the initials climate-weather states, the matrix of probabilities of the climate-weather change process transitions between the climate-weather states, the matrix of conditional distribution functions and the matrix of conditional density functions of the climate-weather change process conditional sojourn times at the climate-weather states are defined. Preliminary applications of the proposed model to the climate-weather change process for the port oil transportation system operation area are presented.

14

Identification methods and procedures of critical infrastructure operation process including operating environment threats

Kołowrocki K., Soszyńska-Budny J.

Journal of Polish Safety and Reliability Association

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2016

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Vol. 7, No. 3

155--168

EN

In the paper, there are presented the methods for identification of the critical infrastructure operation process on the basis of statistical data coming from this process realizations related to the critical infrastructure operating environment threats. These are the methods and procedures for estimating the unknown basic parameters of the critical infrastructure operation process semi-Markov model and identifying the distributions of the critical infrastructure operation process conditional critical infrastructure operation process sojourn times at the particular operation states. There are given the formulae estimating the probabilities of the critical infrastructure operation process straying at the particular operation states at the initial moment, the probabilities of the critical infrastructure operation process transitions between the operation states. Moreover, there are given formulae for the estimator of unknown parameters of the distributions suitable and typical for the description of the critical infrastructure operation process conditional sojourn times at the operation states. Namely, the parameters of the uniform distribution, the triangular distribution, the double trapezium distribution, the quasi-trapezium distribution, the exponential distribution, the Weibull’s distribution and the chimney distribution are estimated using the statistical methods such as the method of moments and the maximum likelihood method. The chi-square goodness-of-fit test is described and proposed to be applied to verifying the hypotheses about these distributions choice validity. The procedure of statistical data sets uniformity analysis based on Kolmogorov-Smirnov test is proposed to be applied to the empirical conditional sojourn times at the operation states coming from different realizations of the same critical infrastructure operation process.

15

Identification methods and procedures of climate-weather change process including extreme weather hazards of port oil piping transportation system operating at land Baltic seaside area

Jakusik E., Kołowrocki K., Kuligowska E., Soszyńska-Budny J., Torbicki M.

Journal of Polish Safety and Reliability Association

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2016

|

Vol. 7, No. 3

57--64

EN

There are presented the methods of identification of the climate-weather change process. These are the methods and procedures for estimating the unknown basic parameters of the climate-weather change process semi-Markov model and identifying the distributions of the climate-weather change process conditional sojourn times at the climate-weather states. There are given the formulae estimating the probabilities of the climate-weather change process staying at the particular climate-weather states at the initial moment, the probabilities of the climate-weather change transitions between the climate-weather states and the parameters of the distributions suitable and typical for the description of the climate-weather change process conditional sojourn times at the particular climate-weather states. The proposed statistical methods applications for the unknown parameters identification of the climate-weather change process model determining the climate-weather change process parameters for the port oil piping transportation system operating at land Baltic seaside area are presented.

16

Identification methods and procedures of climate-weather change process including extreme weather hazards for maritime ferry operating at Baltic Sea open waters

Jakusik E., Kołowrocki K., Kuligowska E., Soszyńska-Budny J., Torbicki M.

Journal of Polish Safety and Reliability Association

|

2016

|

Vol. 7, No. 3

73--80

EN

There are presented the methods of identification of the climate-weather change process. These are the methods and procedures for estimating the unknown basic parameters of the climate-weather change process semi-Markov model and identifying the distributions of the climate-weather change process conditional sojourn times at the climate-weather states. There are given the formulae estimating the probabilities of the climate-weather change process staying at the particular climate-weather states at the initial moment, the probabilities of the climate-weather change transitions between the climate-weather states and the parameters of the distributions suitable and typical for the description of the climate-weather change process conditional sojourn times at the particular climate-weather states. The proposed statistical methods applications for the unknown parameters identification of the climate-weather change process model determining the climate-weather change process parameters for maritime ferry operating at Baltic Sea open waters are presented.

17

Identification methods and procedures of climate-weather change process including extreme weather hazards for the maritime ferry operating at Gdynia port area

Jakusik E., Kołowrocki K., Kuligowska E., Soszyńska-Budny J., Torbicki M.

Journal of Polish Safety and Reliability Association

|

2016

|

Vol. 7, No. 3

65--72

EN

There are presented the methods of identification of the climate-weather change process. These are the methods and procedures for estimating the unknown basic parameters of the climate-weather change process semi-Markov model and identifying the distributions of the climate-weather change process conditional sojourn times at the climate-weather states. There are given the formulae estimating the probabilities of the climate-weather change process staying at the particular climate-weather states at the initial moment, the probabilities of the climate-weather change transitions between the climate-weather states and the parameters of the distributions suitable and typical for the description of the climate-weather change process conditional sojourn times at the particular climate-weather states. The proposed statistical methods applications for the unknown parameters identification of the climate-weather change process model determining the climate-weather change process parameters for the maritime ferry operating at Gdynia port area are presented.

18

Identification methods and procedures of climate-weather change process including extreme weather hazards for port oil piping transportation system operating at under water Baltic Sea area

Jakusik E., Kołowrocki K., Kuligowska E., Soszyńska-Budny J., Torbicki M.

Journal of Polish Safety and Reliability Association

|

2016

|

Vol. 7, No. 3

47--56

EN

There are presented the methods of identification of the climate-weather change process. These are the methods and procedures for estimating the unknown basic parameters of the climate-weather change process semi-Markov model and identifying the distributions of the climate-weather change process conditional sojourn times at the climate-weather states. There are given the formulae estimating the probabilities of the climate-weather change process staying at the particular climate-weather states at the initial moment, the probabilities of the climate-weather change transitions between the climate-weather states and the parameters of the distributions suitable and typical for the description of the climate-weather change process conditional sojourn times at the particular climate-weather states. The proposed statistical methods applications for the unknown parameters identification of the climate-weather change process model determining the climate-weather change process parameters for the initial point of the port oil piping transportation system are presented.

19

Reliability electrical power system of hospital as cold standby system

Grabski F.

Journal of KONBiN

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2016

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No. 1 (37)

153--167

EN

The probabilistic model of a hospital electrical power system consisting of mains, an emergency power system and the automatic transfer switch with the generator starter are discussed in this paper. The reliability model is semi-Markov process describing two different units renewable cold standby system and switch. The embedded Semi-Markov processes concept is applied for description of the system evolution. Time to failure of the system is represented by a random variable denoting the first passage time of the process from the given state to the subset of states. The appropriate theorems of the Semi-Markov processes theory allow us to evaluate the reliability function and some reliability characteristics.

PL

W pracy został przedstawiony model niezawodnościowy sytemu elektroenergetycznego szpitala złożony z podsystemu zasilania sieciowego, podsystemu zasilania awaryjnego oraz automatycznego przełącznika. Niezawodnościowym modelem funkcjonowania systemu jest proces semi-markowski. Model ten jest modyfikacją modelu niezawodności opisującego funkcjonowanie sytemu z rezerwą zimną złożonego z dwóch różnych podsystemów i przełącznika. Do konstrukcji modelu został wykorzystany tak zwany włożony proces semi-markowski. Czas zdatności systemu jest reprezentowany przez czas pierwszego przejścia procesu do określonego podzbioru stanów.

20

Semi-Markov Model of the Operation and Maintenance Process of City Buses

Szubartowski M.

Archives of Transport

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2013

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Vol. 25-26, iss. 1-2

109--116

EN

This paper presents a model of the operation and maintenance process of the technical objects within the urban transport bus system. All the investigations have been presented on the basis of the selected real operation and maintenance system of the city buses. Basing on the identification of the system under investigation and on the process being realized within it a model of the operation and maintenance process of the city buses was built, assuming that the model of this process is the homogeneous semi-Markov process X(t). For this purpose, crucial states of operation and maintenance process in selected transport system were determined as well as possible transfers between those states. Based on this, an event-based model of the operation and maintenance process of the city buses was built, assuming that its model is the homogenous semi-Markov process. For operation data obtained after research conducted in an authentic transport system, values of unconditional periods of duration of process states, values of stationary distribution included in the Markov chain as well as values of probabilities of limit distribution of the semi-Markov process were determined. Based on this, an analysis of the city buses operation and maintenance process in question was performed. Presented in this article the semi-Markov model operation and maintenance process of city buses is the first stage of the creation of the availability model of the urban transport bus system.

PL

W artykule przedstawiono model procesu eksploatacji obiektów technicznych w systemie autobusowej komunikacji miejskiej. Całość rozważań przedstawiono na przykładzie wybranego rzeczywistego systemu eksploatacji autobusów miejskich. Na podstawie identyfikacji badanego systemu i realizowanego w nim procesu zbudowano model procesu eksploatacji autobusów miejskich, zakładając, że modelem tego procesu jest jednorodny proces semi-Markowa X(t). W tym celu wyznaczono istotne stany procesu eksploatacji realizowanego w wybranym systemie transportowym oraz możliwe przejścia między wyróżnionymi stanami. Na tej podstawie zbudowano zdarzeniowy model procesu eksploatacji autobusów miejskich, a następnie matematyczny model tego procesu, zakładając, że jego modelem jest jednorodny proces semi-Markowa. Dla danych eksploatacyjnych, uzyskanych z badań przeprowadzonych w rzeczywistym systemie transportowym, wyznaczono wartości bezwarunkowych czasów trwania stanów procesu, wartości rozkładu stacjonarnego włożonego w proces łańcucha Markowa oraz wartości prawdopodobieństw rozkładu granicznego procesu semi-Markowa. Na tej podstawie dokonano analizy rozpatrywanego procesu eksploatacji autobusów miejskich. Prezentowany w artykule semimarkowski model procesu eksploatacji autobusów miejskich jest pierwszym etapem budowy modelu oceny gotowości systemu autobusowej komunikacji miejskiej.