Method of testing the readiness of means of transport with the use of semi-Markov processes

    Anna Borucka Affiliation


In the analysis of the readiness of means of transport, the Markov and semi-Markov processes are particularly applicable, allowing for the description of the usage process over long periods of time, determination of indicators of the exploitability and readiness of the used set of objects, as well as simulation of long-term forecasts of the usage process results. The studies presented in the literature usually concern the theoretical side of the matter, mainly the construction of formal models of the process of changing the operating states of a vehicle. Less attention is paid to the empirical side, especially with regard to the actual conditions of use. Examples of experimental observations presented in the literature most often concern individual cases. This paper lists selected irregularities and presents an example of a study of a real transport system based on semi-Markov processes.

Keyword : semi-Markov model, readiness, reliability, transport enterprise, means of transport, Markov property, variable distribution

How to Cite
Borucka, A. (2021). Method of testing the readiness of means of transport with the use of semi-Markov processes. Transport, 36(1), 75-83.
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Mar 30, 2021
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Alam, M.; Al-Saggaf, U. M. 1986. Quantitative reliability evaluation of repairable phased-mission systems using Markov approach, IEEE Transactions on Reliability 35(5): 498–503.

Asmussen, S.; Lipsky, L.; Thompson, S. 2016. Markov renewal methods in restart problems in complex systems, in M. Podolskij, R. Stelzer, S. Thorbjornsen, A. Veraart (Eds.). The Fascination of Probability, Statistics and their Applications, 501–527.

Borucka, A.; Niewczas, A.; Hasilova, K. 2019. Forecasting the readiness of special vehicles using the semi-Markov model, Eksploatacja i Niezawodność – Maintenance and Reliability 21(4): 662–669.

Buchholz, P.; Dohndorf, I.; Scheftelowitsch, D. 2018. Time-based maintenance models under uncertainty, Lecture Notes in Computer Science 10740: 3–18.

Bunks, C.; McCarthy, D.; Al-Ani, T. 2000. Condition-based maintenance of machines using hidden Markov models, Mechanical Systems and Signal Processing 14(4): 597–612.

Chen, D.; Trivedi, K. S. 2005. Optimization for condition-based maintenance with semi-Markov decision process, Reliability Engineering & System Safety 90(1): 25–29.

Darong, H.; Lanyan, K.; Xiaoyan, C.; Ling, Z.; Bo, M. 2018. Fault diagnosis for the motor drive system of urban transit based on improved hidden Markov model, Microelectronics Reliability 82: 179–189.

Duan, C.; Makis, V.; Deng, C. 2019. Optimal Bayesian early fault detection for CNC equipment using hidden semi-Markov process, Mechanical Systems and Signal Processing 122: 290–306.

Girtler, J. 2012. Possibility of application of the theory of semi-Markov processes to determine reliability of diagnosing systems, Journal of KONBiN 24(1): 49–58.

Grabski, F. 2015. Semi-Markov Processes: Applications in System Reliability and Maintenance. Elsevier. 270 p.

Grabski, F. 2017. Semi-Markov reliability model of two different units cold standby system, Zeszyty Naukowe Akademii Marynarki Wojennej – Scientific Journal of Polish Naval Academy 58(4): 45–60.

Grabski, F.; Jaźwiński, J. 2009. Funkcje o losowych argumentach w zagadnieniach niezawodności, bezpieczeństwa i logistyki. Wydawnictwa Komunikacji i Łączności. 344 s. (in Polish).

Gupta, R.; Tyagi, A. 2019. A discrete parametric Markov-chain model of a two-unit cold standby system with repair efficiency depending on environment, Reliability: Theory & Applications – RT&A 14(1): 23–33.

Hunter, J. J. 2016. Accurate calculations of stationary distributions and mean first passage times in Markov renewal processes and Markov chains, Special Matrices 4(1): 151–175.

Knopik, L.; Migawa, K. 2017. Optimal age-replacement policy for non-repairable technical objects with warranty, Eksploatacja i Niezawodność – Maintenance and Reliability 19(2): 172–178.

Kozłowski, E.; Borucka, A.; Świderski, A. 2020. Application of the logistic regression for determining transition probability matrix of operating states in the transport systems, Eksploatacja i Niezawodność – Maintenance and Reliability 22(2): 192–200.

Kozłowski, E.; Mazurkiewicz, D.; Żabiński, T.; Prucnal, S.; Sęp, J. 2019. Assessment model of cutting tool condition for realtime supervision system, Eksploatacja i Niezawodność – Maintenance and Reliability 21(4): 679–685.

Landowski, B.; Muślewski, Ł.; Knopik, L.; Bojar, P. 2017. Semi-Markov model of quality state changes of a selected transport system, Journal of KONES 24(4): 141–148.

Li, Y.; Dong, Y.-N.; Zhang, H.; Zhao, H.-T.; Shi, H.-X.; Zhao, X.-X. 2010. Spectrum usage prediction based on high-order Markov model for cognitive radio networks, in 2010 10th IEEE International Conference on Computer and Information Technology, 29 June – 1 July 2010, Bradford, UK, 2784–2788.

Limnios, N.; Oprişan, G. 2001. Semi-Markov Processes and Reliability. Springer Science + Business Media New York. 222 p.

Lisnianski, A.; Frenkel, I. 2009. Non-homogeneous Markov reward model for aging multi-state system under minimal repair, International Journal of Performability Engineering 5(4): 303–312.

Love, C. E.; Zhang, Z. G.; Zitron, M. A.; Guo, R. 2000. A discrete semi-Markov decision model to determine the optimal repair/replacement policy under general repairs, European Journal of Operational Research 125(2): 398–409.

Papamichail, C. A.; Bouzebda, S.; Limnios, N. 2016. Reliability calculus on crack propagation problem with a Markov renewal process, Computational Methods in Applied Sciences 41: 343–378.

Perman, M.; Senegacnik, A.; Tuma, M. 1997. Semi-Markov models with an application to power-plant reliability analysis, IEEE Transactions on Reliability 46(4): 526–532.

Restel, F. J. 2014. The Markov reliability and safety model of the railway transportation system, in T. Nowakowski, M. Mlynczak, A. Jodejko-Pietruczuk, S. Werbinska-Wojciechowska (Eds.). Safety and Reliability: Methodology and Applications, 303–311.

Rudnicki, J. 2011. The time of the first transition of the semi-Markov process in the evaluation of diesel engine operation, Silniki Spalinowe – Combustion Engines 50(2): 89–98.

Rydén, T.; Teräsvirta, T.; Asbrink, S. 1998. Stylized facts of daily return series and the hidden Markov model, Journal of Applied Econometrics 13(3): 217–244.<217::AID-JAE476>3.0.CO;2-V

Rymarczyk, T.; Kozłowski, E.; Kłosowski, G.; Niderla, K. 2019. Logistic regression for machine learning in process tomography, Sensors 19(15): 3400.

Rymarz, J.; Niewczas, A.; Krzyżak, A. 2016. Comparison of operational availability of public city buses by analysis of variance, Eksploatacja i Niezawodność – Maintenance and Reliability 18(3): 373–378.

Sanusi, W.; Jemain, A. A.; Zin, W. Z. W.; Zahari, M. 2015.The drought characteristics using the first-order homogeneous Markov chain of monthly rainfall data in Peninsular Malaysia, Water Resources Management 29(5): 1523–1539.

Shi, S.; Lin, N.; Zhang, Y.; Huang, C.; Liu, L.; Lu, B.; Cheng, J. 2013. Research on Markov property analysis of driving cycle, in 2013 IEEE Vehicle Power and Propulsion Conference (VPPC), 15–18 October 2013, Beijing, China, 1–5.

Tang, D.; Sheng, W.; Yu, J. 2018. Dynamic condition-based maintenance policy for degrading systems described by a randomcoefficient autoregressive model: a comparative study, Eksploatacja i Niezawodność – Maintenance and Reliability 20(4): 590–601.

The R Foundation. 2019. The R Project for Statistical Computing. The R Foundation for Statistical Computing, Vienna, Austria. Available from Internet:

Van Casteren, J. F. L.; Bollen M. H. J.; Schmieg M. E. 2000. Reliability assessment in electrical power systems: the Weibull–Markov stochastic model, IEEE Transactions on Industry Application 36(3): 911–915.

Wang, Y.; Infield, D. 2018. Markov chain Monte Carlo simulation of electric vehicle use for network integration studies, International Journal of Electrical Power & Energy Systems 99: 85–94.

Wang, Z.-Z.; Huang, X.; Liang, Y.-R. 2018. Oil-gas reservoir lithofacies stochastic modeling based on one- to three-dimensional Markov chains, Journal of Central South University 25(6): 1399–1408.

Woropay, M.; Landowski, B.; Neubauer A. 2004. Using of decision-making semi-Markov processes for modelling and simulating of operation and maintenance process for buses, The Archives of Automotive Engineering – Archiwum Motoryzacji 7(1): 53–65.

Wu, B.; Cui, L.; Fang, C. 2019. Reliability analysis of semi-Markov systems with restriction on transition times, Reliability Engineering & System Safety 190: 106516.

Zhang, Y.-F.; Zhang, Q.-F.; Yu, R.-H. 2010. Markov property of Markov chains and its test, in 2010 International Conference on Machine Learning and Cybernetics, 11–14 July 2010, Qingdao, China, 1864–1867.

Zhang, Z.; Zhou, Y.; Sun, Y.; Ma, L. 2012. Condition-based maintenance optimisation without a predetermined strategy structure for a two-component series system, Eksploatacja I Niezawodność – Maintenance and Reliability 14(2): 120–129.

Żurek, J.; Tomaszewska, J. 2016. Analiza systemu eksploatacji z punktu widzenia gotowości, Prace Naukowe Politechniki Warszawskiej 114: 471–477. (in Polish).