Skip to main content
SpringerLink
Log in

Tight Bounds on the Coefficients of Consecutive k-out-of-n:F Systems

  • Conference paper
  • First Online:
Intelligent Methods in Computing, Communications and Control (ICCCC 2020)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1243))

Included in the following conference series:

Abstract

In this paper we compute the coefficients of the reliability polynomial of a consecutive-k-out-of-n:F system, in Bernstein basis, using the generalized Pascal coefficients. Based on well-known combinatorial properties of the generalized Pascal triangle we determine simple closed formulae for the reliability polynomial of a consecutive system for particular ranges of k. Moreover, for the remaining ranges of k (where we were not able to determine simple closed formulae), we establish easy to calculate sharp bounds for the reliability polynomial of a consecutive system.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    John von Neumann presented his work in five seminal lectures at the California Institute of Technology (Caltech) in January 1952. They are available, based on the notes taken by R. S. Pierce, at https://sites.google.com/site/michaeldgodfrey/vonneumann/vN_Caltech_Lecture.pdf.

References

  1. Beichl, I., Cloteaux, B., Sullivan, F.: An approximation algorithm for the coefficients of the reliability polynomial. Congr. Numer. 197, 143–151 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Camarda, P.: Bounds evaluation of coefficients in the reliability polynomial. Microelectr. Reliab. 30(6), 1099–1110 (1990)

    Article  Google Scholar 

  3. Chang, G.J., Cui, L., Hwang, F.K.: Reliabilities of Consecutive-k Systems. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  4. Chari, M., Colbourn, C.J.: Reliability polynomials: a survey. J. Combin. Inform. System Sci. 22(3–4), 177–193 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Chiang, D.T., Niu, S.: Reliability of consecutive-\(k\)-out-of-\(n\):\({F}\) system. IEEE Trans. Reliab. R–30(1), 87–89 (1981)

    Article  Google Scholar 

  6. Cluzeau, T., Keller, J., Schneeweiss, W.: An efficient algorithm for computing the reliability of consecutive-\(k\)-out-of-\(n\):\({F}\) systems. IEEE Trans. Reliab. 57(1), 84–87 (2008)

    Article  Google Scholar 

  7. Colbourn, C.J.: The Combinatorics of Network Reliability. Oxford University Press, New York (1987)

    Google Scholar 

  8. Daus, L., Beiu, V.: Lower and upper reliability bounds for consecutive-\(k \)-out-of-\(n\):\({F}\) systems. IEEE Trans. Reliab. 64(3), 1128–1135 (2015)

    Article  Google Scholar 

  9. de Moivre, A.: The Doctrine of Chances. W. Pearson, London (1971). 3rd edn. (1756). https://archive.org/details/doctrineofchance00moiv/

  10. Deng, H., Chen, J., Li, Q., Li, R., Gao, Q.: On the construction of most reliable networks. Discr. Appl. Maths. 140(1–3), 19–33 (2004)

    Article  MathSciNet  Google Scholar 

  11. Eryilmaz, S.: Review of recent advances in reliability of consecutive-\(k\)-out-of-\(n\) and related systems. J. Risk Reliab. 224(3), 225–237 (2010)

    Google Scholar 

  12. Gokdere, G., Gurkan, M., Kilic, M.B.: A new method for computing the reliability of consecutive \(k\)-out-of-\(n\):\({F}\) systems. Open Phys. 14(1), 166–170 (2016)

    Article  Google Scholar 

  13. Kontoleon, J.M.: Reliability determination of a \(r\)-successive-out-of-\(n\):\({F}\) system. IEEE Trans. Reliab. R–29(5), 437 (1980)

    Article  Google Scholar 

  14. Mohammadi, F., Sáenz-de Cabezón, E., Wynn, H.P.: Efficient multicut enumeration of \(k\)-out-of-\(n\):\({F}\) and consecutive \(k\)-out-of-\(n\):\({F}\) systems. Pattern Recognit. Lett. 102(1), 82–88 (2018)

    Article  Google Scholar 

  15. Moore, E.F., Shannon, C.E.: Reliable circuits using less reliable relays - Part I. J. Frankl. Inst. 262(3), 191–208 (1956)

    Article  Google Scholar 

  16. Oxley, J., Welsh, D.: Chromatic, flow and reliability polynomials: the complexity of their coefficients. Comb. Probab. Comput. 11(4), 403–426 (2002)

    Article  MathSciNet  Google Scholar 

  17. Pérez-Rosés, H.: Sixty years of network reliability. Maths. Comp. Sci. 12(3), 275–293 (2018)

    Article  MathSciNet  Google Scholar 

  18. Provan, J.S.: Bounds on the reliability of networks. IEEE Trans. Reliab. R–35(3), 260–268 (1986)

    Article  MathSciNet  Google Scholar 

  19. Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12(4), 777–788 (1983)

    Article  MathSciNet  Google Scholar 

  20. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)

    Article  MathSciNet  Google Scholar 

  21. von Neumann, J.: Probabilistic logics and the synthesis of reliable organisms from unreliable components. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 43–98. Princeton University Press, Princeton (1956)

    Google Scholar 

Download references

Acknowledgements

This research was supported by the European Union through the European Regional Development Fund under the Competitiveness Operational Program (BioCell-NanoART = Novel Bio-inspired Cellular Nano-Architectures, POC-A1.1.4-E-2015 nr. 30/01.09.2016).

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, “Aurel Vlaicu” University of Arad, Arad, Romania

    Vlad-Florin Drăgoi, Simon Cowell & Valeriu Beiu

  2. LITIS, University of Rouen Normandie Avenue de l’université, 76801, Saint-Étienne-du-Rouvray, France

    Vlad-Florin Drăgoi

Authors
  1. Vlad-Florin Drăgoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon Cowell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valeriu Beiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vlad-Florin Drăgoi .

Editor information

Editors and Affiliations

  1. Agora University of Oradea, Oradea, Romania

    Ioan Dzitac

  2. University of Oradea, Oradea, Romania

    Simona Dzitac

  3. Academia Romana, Bucharest, Romania

    Florin Gheorghe Filip

  4. Polish Academy of Sciences, Systems Research Institute, Warszawa, Poland

    Janusz Kacprzyk

  5. Agora University of Oradea, Oradea, Romania

    Misu-Jan Manolescu

  6. University of Oradea, Oradea, Romania

    Horea Oros

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Drăgoi, VF., Cowell, S., Beiu, V. (2021). Tight Bounds on the Coefficients of Consecutive k-out-of-n:F Systems. In: Dzitac, I., Dzitac, S., Filip, F., Kacprzyk, J., Manolescu, MJ., Oros, H. (eds) Intelligent Methods in Computing, Communications and Control. ICCCC 2020. Advances in Intelligent Systems and Computing, vol 1243. Springer, Cham. https://doi.org/10.1007/978-3-030-53651-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-53651-0_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-53650-3

  • Online ISBN: 978-3-030-53651-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation