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On an Interrupted Bivariate Renewal Process and Its Applications

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Abstract

A special case of the bivariate renewal process is investigated. It is supposed, that this process is considered while the second component has a positive value. The algorithm for a calculation of the corresponding time’s density is presented. In addition, a case of preventive renewal is considered. Such renewal takes place when the value of the second component is positive but is less than a fixed level. The following characteristics are investigated: distribution of the number of such renewals, the density of the time of the failure, etc. Numerical examples illustrate the given presentation.

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REFERENCES

  1. Cox, D.R., Renewal Theory, London: Methuen & Co, 1962.

    MATH  Google Scholar 

  2. Smith, W.L., Renewal theory and its ramifications, J. R. Stat. Soc.: Ser. B (Methodological), 1958, vol. 20, no. 2, pp. 243–284. https://doi.org/10.1111/j.2517-6161.1958.tb00294.x

    Article  MathSciNet  MATH  Google Scholar 

  3. Bickel, P.J. and Yahav, J.A., Renewal theory in the plane, Ann. Math. Stat., 1985, vol. 36, no. 3, pp. 946–955. https://doi.org/10.1214/aoms/1177700067

    Article  MathSciNet  MATH  Google Scholar 

  4. Hunter, J.J., Renewal theory in two dimensions: Basic results, Adv. Appl. Probab., 1974, vol. 6, no. 2, pp. 376–391. https://doi.org/10.2307/1426299

    Article  MathSciNet  MATH  Google Scholar 

  5. Arunachalam, V. and Calvache, Á., Approximation of the bivariate renewal function, Commun. Stat. Simul. Comput., 2015, vol. 44, no. 1, pp. 154–167. https://doi.org/10.1080/03610918.2013.770306

    Article  MathSciNet  MATH  Google Scholar 

  6. Eliashberg, J., Singpurwalla, N.D., and Wilson, S.P., Calculating the reserve for a time and usage indexed warranty, Manage. Sci., 1997, vol. 43, no. 7, pp. 966–975. https://doi.org/10.1287/mnsc.43.7.966

    Article  MATH  Google Scholar 

  7. Hadji, E.M., Kambo, N.S., and Rangan, A., Two-dimensional renewal function approximation, Commun. Stat. Theory Methods, 2015, vol. 44, no. 15, pp. 3107–3124. https://doi.org/10.1080/03610926.2013.815204

    Article  MathSciNet  MATH  Google Scholar 

  8. Hunter, J.J., Renewal theory in two dimensions: Asymptotic results, Adv. Appl. Probab., 1974, vol. 6, no. 3, pp. 546–562. https://doi.org/10.2307/1426233

    Article  MathSciNet  MATH  Google Scholar 

  9. Mitov, K.V. and Omey, E., Intuitive approximations for the renewal function, Stat. Probab. Lett., 2015, vol. 84, pp. 72–80. https://doi.org/10.1016/j.spl.2013.09.030

    Article  MathSciNet  MATH  Google Scholar 

  10. Omey, E., Mitov, K.V., and Vesilo, R., Approximations in bivariate renewal theory, Publ. Inst. Math. (Belgrade), 2018, vol. 104, no. 118, pp. 69–88. https://doi.org/10.2298/pim1818069o

    Article  MathSciNet  MATH  Google Scholar 

  11. Chung, K.L., On the renewal theorem in higher dimensions, Skand. Aktuarial J., 1952, vol. 35, nos. 3–4, pp. 188–194.

    MathSciNet  MATH  Google Scholar 

  12. Kaniskauskas, V. and Dronova-Plartbardze, L., The renewal equation for multivariate renewal processes, Siauliai Math. Seminar, 2010, vol. 5, no. 13, pp. 47–53.

  13. Spitzer, F., A multidimensional renewal theorem, Probability, Statistical Mechanics, and Number Theory, Rota, G.-C., Ed., Advances in Mathematics Supplemental Studies, vol. 9, Orlando, Fla.: Academic, 1986, pp. 147–155.

  14. Steinebach, J. and Eastwood, V.R., Extreme value asymptotics for multivariate renewal processes, J. Multivariate Anal., 1996, vol. 56, no. 2, pp. 284–302. https://doi.org/10.1006/jmva.1996.0015

    Article  MathSciNet  MATH  Google Scholar 

  15. Gertsbakh, I.B., Models of Preventive Maintenance, Studies in Mathematical and Managerial Economics, vol. 23, Oxford: North-Holland Publishing, 1977.

  16. Stadje, W. and Zuckerman, D., Optimal maintenance strategies for repairable systems with general degree of repair, J. Appl. Probab., 1991, vol. 28, no. 2, pp. 384–396. https://doi.org/10.2307/3214874

    Article  MathSciNet  MATH  Google Scholar 

  17. Stadje, W. and Zuckerman, D., Optimal strategies for some repair replacement models, Adv. Appl. Probab., 1990, vol. 22, no. 3, pp. 641–656. https://doi.org/10.2307/1427462

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, S.-C., A bivariate renewal process and its applications in maintenance policies, PhD Dissertation, Blacksburg, Va.: Virginia Polytechnic Institute and State University, 1999.

  19. Yang, S.-Ch. and Nachlass, J.A., Bivariate reliability and availability modeling, IEEE Trans. Reliab., 2001, vol. 50, no. 1, pp. 26–35. https://doi.org/10.1109/24.935013

    Article  Google Scholar 

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Authors and Affiliations

  1. Transport and Telecommunication Institute, Riga, Latvia

    A. Andronov

  2. UiT The Arctic University of Norway, Faculty of Engineering Science and Technology, Narvik, Norway

    D. Santalova Thordarson & Hao Yu

Authors
  1. A. Andronov
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  2. D. Santalova Thordarson
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  3. Hao Yu
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Corresponding authors

Correspondence to A. Andronov, D. Santalova Thordarson or Hao Yu.

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Andronov, A., Thordarson, D.S. & Yu, H. On an Interrupted Bivariate Renewal Process and Its Applications. Aut. Control Comp. Sci. 57, 490–503 (2023). https://doi.org/10.3103/S0146411623050036

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  • DOI: https://doi.org/10.3103/S0146411623050036

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