Abstract
In the present paper, we study the Birnbaum’s Reliability Importance measures of the components for Markov Chain Imbeddable structures. A generating function approach is presented for determining the Birnbaum’s importance measure for a reliability system consisting of independent and identically distributed components. The general setup is implemented for two specific members of the class of Markov Chain Imbeddable structures. The resulting recurrences for computing the corresponding important measures are established. More precisely, the consecutive-k-out-of-n: G system and the k-out-of-n: G system are under investigation. For the first-mentioned structure a recursive scheme (accompanied with the necessary set of initial conditions) for the Birnbaum’s importance measure of its components is introduced, while for the latter one an explicit formula for determining the corresponding quantities is proved. A numerical investigation is carried out by the aid of the theoretical outcomes of the manuscript and some concluding remarks about the design parameters of the underlying systems are also discussed.
Similar content being viewed by others
References
Andreao RV, Dorizzi B, Boudy J (2006) ECG signal analysis through hidden Markov models. IEEE Trans Biomed Eng 53:1541–1549
Birnbaum ZW (1969) On the importance of different components in a multicomponent system. In: Krishnaiah PR (ed) Multivariate analysis II. Academic, New York, pp 581–592
Bisht S, Kumar A, Goyal N, Ram M, Klochkov Y (2021) Analysis of network reliability and importance of components in a communication network. Mathematics 9:1347
Boland PJ, El-Neweihi E (1995) Measures of component importance in reliability theory. Comput Oper Res 22:455–463
Chadjiconstantinidis S, Koutras MV (1999) Measures of component importance for Markov chain imbeddable reliability structures. Nav Res Logist 46:613–639
Chang JG, Cui L, Hwang FK (2000) Reliabilities of consecutive-k systems. Kluwer Academic Publishers, Dordrecht
Chao MT, Fu JC (1991) The reliability of large series systems under Markov structure. Adv Appl Probab 23:894–908
Chao MT, Fu JC, Koutras MV (1995) Survey of reliability studies of consecutive-k-out-of-n: F & related systems. IEEE Trans Reliab 44(1):120–127
Chen J-C, Wu YJ (2020) Discrete-time Markov chain for prediction of air quality index. J Ambient Intell Humaniz Comput. https://doi.org/10.1007/s12652-020-02036-5
D’Arienzo MP, Dudin AN, Dudin SA, Munzo R (2020) Analysis of a retrial queue with group service of impatient customers. J Ambient Intell Humaniz Comput 11:2591–2599
Eryilmaz S (2010) Review of recent advances in reliability of consecutive-k-out-of-n: F and related systems. Proc Instit Mech Eng-Part O- J Risk Reliab 224(3):225–237
Ferrández-Pastor FJ, Mora-Mora H, Sánchez-Romero JL, Nieto-Hidalgo M, García-Chamizo JM (2017) Interpreting human activity from electrical consumption data using reconfigurable hardware and hidden Markov models. J Ambient Intell Humaniz Comput 8:469–483
Fu JC, Koutras MV (1994) Distribution theory of runs: a Markov chain approach. J Am Stat Assoc 89:957–974
Gupta A, Srinath P (2022) A recommender system based on collaborative filtering, graph theory using HMM based similarity measures. Int J Syst Assur Eng Manag 13:533–545
Huseby AB, Kalinowska M, Abrahamsen T (2022) Birnbaum criticality and importance measures for multistate systems with repairable components. Probab Eng Inf Sci 36:66–86
Koutras MV (1996) Runs, scans and urn models: a unified Markov chain approach. Ann Inst Stat Math 47:743–766
Koutras MV, Alexandrou VA (1995) On a Markov chain approach for the study of reliability Structures. J Appl Probab 33:357–367
Kumar A, Ram M (2017) Effects of k-out-of-n:G/F and parallel redundancy in an industrial system through reliability approach. Proc AIP Conf 1860:020046
Kumar G, Jain V, Gandhi OP (2014) Steady-state availability analysis of repairable mechanical systems with opportunistic maintenance by using Semi-Markov process. Int J Syst Assur Eng Manag 5:664–678
Kuo W, Zuo MJ (2003) Optimal reliability modeling: principles and applications. Wiley, New York
Kuo W, Zhang W, Zuo MJ (1990) A consecutive-k-out-of-n: G system: the mirror image of a consecutive-k-out-of-n: F system. IEEE Trans Reliab 39(2):244–253
Levitin G (2004) Consecutive k−out−of−r−from−n system with multiple failure criteria. IEEE Trans Reliab 53:394–400
Lou WYW (1996) On runs and longest run tests-a method of finite Markov chain imbedding. J Am Stat Assoc 91:1595–1601
Miziula P, Navarro J (2019) Birnbaum importance measure for reliability systems with dependent components. IEEE Trans Reliab 68:439–450
Okamura H, Dohi T (2010) Software safety assessment based on a subordinated Markov chain. Int J Syst Assur Eng Manag 1:307–315
Qiu S, Ming Z, Sallak M, Lu J (2022) A Birnbaum importance-based two-stage approach for two-type component assignment problems. Reliab Eng Syst Saf 218:108051
Rahimifar A, Kavian YS, Kaabi H, Soroosh M (2021) Predicting the energy consumption in software defined wireless sensor networks: a probabilistic Markov model approach. J Ambient Intell Humaniz Comput 12:9053–9066
Ram M (2013) On system reliability approaches: a brief survey. Int J Syst Assur Eng Manag 4(2):101–117
Sarje AK, Prasad EV (1989) An efficient non-recursive algorithm for computing the reliability of k-out-of-n systems. IEEE Trans Reliab 38:234–235
Triantafyllou IS (2021) Reliability study of <n, f,2> systems: a generating function approach. Int J Math Eng Manag Sci 6(1):44–65
Triantafyllou IS, Koutras MV (2011) Signature and IFR preservation of 2-within-consecutive-k-out-of-n: F systems and applications. IEEE Trans Reliab 60:315–322
Triantafyllou IS (2015) Consecutive-type reliability systems: an overview and some applications. J Quality Reliab Eng 2015:Article ID 212303
Yousefi HHN, Kavian Y (2022) A Markov chain model for IEEE 802.15.4 in time critical wireless sensor networks under periodic traffic with reneging packets. J Ambient Intell Humaniz Comput 13:2253–2268
Zazhigalkin AV, Aronov IZ, Maksimova OV, Papic L (2019) Control of consensus convergence in technical committees of standardization on the basis of regular Markov chains model. Int J Syst Assur Eng Manag 10:29–36
Zuo MJ, Lin D, Wu Y (2000) Reliability evaluation of combined k−out−of−n: F, consecutive−k−out−of−n: F and linear connected−(r, s)−out−of− (m, n): F system structures. IEEE Trans Reliab 49(1):99–104
Funding
No outside funding or grants directly related to the research presented in this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Research involving Human Participants and/or Animals
This article does not contain any studies with human participants or animals performed by any of the authors.
Informed consent
The author has consented to the submission to the journal.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Triantafyllou, I.S. On the importance measures of Markov chain imbeddable reliability systems: some advances for k-out-of-n: G and consecutive-k-out-of-n: G structures. Int J Syst Assur Eng Manag 15, 1434–1443 (2024). https://doi.org/10.1007/s13198-023-01903-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13198-023-01903-6