Fuzzy semi-Markovian stochastic model for single-unit system with repairman arrival delay under Lindley lifetime distribution using bell-shaped membership function

Authors

  • Monika Saini
    Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur-303007, India
  • Ashish Kumar
    Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur-303007, India

Keywords:

Fuzzy Availability, Lindley Distribution, Single-unit system, Bell-shaped membership function, Regenerative point technique

Abstract

The key objective of the present analysis is to analyze the impact of fuzziness and repairman arrival delay on the mean time to system failure (MTSF) and availability of single-unit systems. Using a semi-Markovian approach, two stochastic models were developed, and recursive relations were obtained for the reliability measures. The lifetime, failure time, and repairman arrival time are considered Lindley-distributed random variables, with a parameter represented as a fuzzy number defined by a bell-shaped membership function. The switching devices and repairs are assumed to be as good as new. The numerical results show that increasing the \alpha-cut reduces fuzziness and improves the estimates, while immediate arrival of the repairman increases system availability. The importance of the results is highlighted using a numerical example generated from a Lindley-distributed random sample of size 1000 by considering \varepsilon = 0.1 and dynamic \delta in RStudio.

Dimensions

[1] M. K. Loganathan, G. Kumar & O. P. Gandhi, ``Availability evaluation of manufacturing systems using semi-Markov model'', International journal of computer integrated manufacturing 29 (2016) 720. https://doi.org/10.1080/0951192X.2015.1068454.

[2] Gitanjali, ``Reliability measurement of complex industrial redundant systems using semi-Markov process and regenerative point technique'', in recent advances in metrology: select proceedings of AdMet 2021, Springer, Singapore, 2023, pp. 221--231. https://doi.org/10.1007/978-981-19-2468-2_25.

[3] F. Grabski, ``Semi-Markov reliability model of two different units cold standby system'', Zeszyty Naukowe akademii Marynarki Wojennej 58 (2017) 45. https://scispace.com/pdf/semi-markov-reliability-model-of-two-different-units-cold-54q7fgntgw.pdf.

[4] Y. Rani, I. Kumar & Gitanjali, ``Reliability modeling of parallel tubular and plate heat exchangers using regenerative point technique'', International journal of information technology 17 (2025) 1. https://doi.org/10.1007/s41870-025-02636-4.

[5] A. Kumar, M. Saini & D. K. Srivastava, ``Performance analysis of a cold standby system with fault detection and arrival time of server subject to imperfect coverage'', International journal of mathematics and its applications 4 (2016) 139. https://ijmaa.in/index.php/ijmaa/article/view/554.

[6] V. Singhal & I. Kumar, ``Enhancing water supply continuity: a semi-Markovian reliability assessment of multi-pump treatment systems'', International journal of information technology (2025). https://doi.org/10.1007/s41870-025-02788-3.

[7] N. Nandal & S. C. Malik, ``On use of gamma distribution for evaluation of reliability and availability of a single unit system subject to arrival time of the server'', Journal of reliability and statistical studies 12 (2019) 93. https://doi.org/10.13052/jrss2229-5666.1228.

[8] N. Nandal, Availability analysis of systems with different configurations and repair policies, Ph.D. dissertation, Maharshi Dayanand University, Rohtak, India, 2020. http://hdl.handle.net/10603/326446.

[9] M. E. Ghitany, B. Atieh & S. Nadarajah, ``Lindley distribution and its application'', Mathematics and computers in simulation 78 (2008) 493. https://doi.org/10.1016/j.matcom.2007.06.007.

[10] S. C. Malik & N. Nandal, ``Availability analysis of a three unit cold standby system using Lindley failure and repair laws'', International journal of agricultural & statistical sciences 15 (2019) 543. https://connectjournals.com/file_full_text/3053802H_543-548.pdf.

[11] D. Qayoom, A. A. Rather, N. Alsadat, E. Hussam & A. M. Gemeay, ``A new class of Lindley distribution: system reliability, simulation and applications'', Heliyon 10 (2024) e38335. https://doi.org/10.1016/j.heliyon.2024.e38335.

[12] M. A. M. Safari, N. Masseran & M. H. Abdul Majid, ``Robust reliability estimation for Lindley distribution: a probability integral transform statistical approach'', Mathematics 8 (2020) 1634. https://doi.org/10.3390/math8091634.

[13] R. Alotaibi & A. Elshahhat, ``A new discrete model of Lindley families: theory, inference, and real-world reliability analysis'', Mathematics 14 (2026) 397. https://doi.org/10.3390/math14030397.

[14] A. Yaghoubi, ``Analysis of repairable systems availability with Lindley failure and repair behavior'', arXiv preprint arXiv:2602.07935 (2026). https://doi.org/10.48550/arXiv.2602.07935.

[15] M. Tyagi & P. K. Poonia, ``Performance evaluation and reliability analysis of a repairable gas station using weighted exponential Lindley distribution'', International journal of quality engineering and technology 11 (2025) 92. https://doi.org/10.1504/IJQET.2025.10075087.

[16] P. S. Pundir & R. Patawa, ``Stochastic behavior of dissimilar units cold standby system waiting for repair'', Life cycle reliability and safety engineering 8 (2019) 43. https://doi.org/10.1007/s41872-018-00070-z.

[17] S. Sharma & V. Kumar, ``Reliability estimation in a two-unit hot standby system under classical and Bayesian inferential framework'', Proceedings of the institution of mechanical engineers, Part O: journal of risk and reliability 240 (2026) 105. https://doi.org/10.1177/1748006X251360273.

[18] S. Varshney, M. Bajaj, K. K. Choudhary, M. Pushkarna & I. Zaitsev, ``Performance analysis and reliability prediction of multi-state service systems with multiple failure modes of unreliable server: an engineering perspective'', Engineering reports 7 (2025) e70268. https://doi.org/10.1002/eng2.70268.

[19] V. S. Maan, M. Saini & A. Kumar, ``Investigation of fuzzy semi-Markovian model for single unit systems with partial failure and Weibull distributed random laws'', International journal of information technology 14 (2022) 2971. https://doi.org/10.1007/s41870-022-01070-0.

[20] M. Savoia, ``Structural reliability analysis through fuzzy number approach, with application to stability'', Computers & structures 80 (2002) 1087. https://doi.org/10.1016/S0045-7949(02)00068-8.

[21] F. Maturo & F. Fortuna, ``Bell-shaped fuzzy numbers associated with the normal curve'', Topics on methodological and applied statistical inference, Springer, Cham, Switzerland, 2017, pp. 131--144. https://doi.org/10.1007/978-3-319-44093-4_13.

[22] P. Dutta & B. Limboo, ``Bell-shaped fuzzy soft sets and their application in medical diagnosis'', Fuzzy information and engineering 9 (2017) 67. https://doi.org/10.1016/j.fiae.2017.03.004.

[23] A. Ranjan & S. K. Dash, ``Bell-shaped fuzzy decision tree: A novel approach for improved decision making'', International journal of advanced technology and engineering exploration 12 (2025) 90. https://accentsjournals.org/PaperDirectory/Journal/IJATEE/2025/1/6.pdf.

[24] L. A. Zadeh, ``Probability measures of fuzzy events'', Journal of mathematical analysis and applications 23 (1968) 421. https://doi.org/10.1016/0022-247X(68)90078-4.

[25] M. S. Kadyan, J. Bura & J. Kumar, ``Application of picture fuzzy set theory to reliability and risk analysis of complex industrial systems'', Life cycle reliability and safety engineering 13 (2024) 293. https://doi.org/10.1007/s41872-024-00262-w.

[26] S. Gupta, P. K. Biswas, B. Aljafari, S. B. Thanikanti & S. K. Das, ``Modelling, simulation and performance comparison of different membership functions based fuzzy logic control for an active magnetic bearing system'', The journal of engineering 2023 (2023) e12229. https://doi.org/10.1049/tje2.12229.

[27] N. S. Y. Temraz, ``Comparison of fuzzy semi-Markov models for one unit with mixed standby units with and without preventive maintenance using regenerative point method'', Heliyon 7 (2021) e07717. https://doi.org/10.1016/j.heliyon.2021.e07717.

[28] M. Saini & A. Kumar, ``Investigation of mean time to system failure of fuzzy semi-Markovian repairable redundant system'', International conference on information and communication technology for intelligent systems, Springer, Singapore, 2024, pp. 431--439. https://doi.org/10.1007/978-981-97-6675-8_35.

[29] Y. Liu, W. Tang & R. Zhao, ``Reliability and mean time to failure of unrepairable systems with fuzzy random lifetimes'', IEEE Transactions on Fuzzy Systems 15 (2007) 1009. https://doi.org/10.1109/TFUZZ.2006.890677.

[30] Y. Zhang & Y. Zhang, ``Reliability analysis of a parallel dependent system with a single cold standby unit'', International journal of industrial and systems engineering 23 (2016) 166. https://doi.org/10.1504/IJISE.2016.076398.

[31] U. Kumari & D. C. Sharma, ``Fuzzy analysis of a retrial machine repair problem using Gaussian fuzzy number'', Palestine Journal of Mathematics 13 (2024) 722. https://pjm.ppu.edu/paper/1813-fuzzy-analysis-retrial-machine-repair-problem-using-gaussian-fuzzy-number.

[32] D. S. Hooda & M. S. Barak, ``Estimation of missing values in fuzzy matrices (FM) and interval-valued fuzzy matrices (IVFM)'', Life cycle reliability and safety engineering 9 (2020) 241. https://doi.org/10.1007/s41872-020-00116-1.

[33] D. S. Hooda & M. S. Barak, ``Estimation of missing data in design of experiment and contingency table'', SN applied sciences 1 (2019) 670. https://doi.org/10.1007/s42452-019-0692-0.

fig2

Published

2026-06-29

How to Cite

Fuzzy semi-Markovian stochastic model for single-unit system with repairman arrival delay under Lindley lifetime distribution using bell-shaped membership function. (2026). Journal of the Nigerian Society of Physical Sciences, 8(3), 3438. https://doi.org/10.46481/jnsps.2026.3438

Issue

Volume 8, Issue 3, August 2026 (In Progress)

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2026 Monika Saini, Ashish Kumar (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Fuzzy semi-Markovian stochastic model for single-unit system with repairman arrival delay under Lindley lifetime distribution using bell-shaped membership function. (2026). Journal of the Nigerian Society of Physical Sciences, 8(3), 3438. https://doi.org/10.46481/jnsps.2026.3438

Similar Articles

11-20 of 275

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)