Skip to main content
SpringerLink
Log in

Studying high reliability systems in the probabilistic school of B. V. Gnedenko

  • Mathematical Models and Methods of Reliability Theory
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We give a survey of the works of B.V. Gnedenko’s reliability school, starting from 1950s and up until the latest years in two directions: (1) invariance of state distributions for queueing systems and networks, (2) asymptotic behavior of a redundant system’s characteristics under low load.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Gnedenko, B.V. and Kovalenko, I.N., Lektsii po teorii massovogo obsluzhivaniya. Vyp. 1–3 (Lectures on Queueing Theory. Issues 1–3), Kiev: KVIRTU, 1963.

    Google Scholar 

  2. Gnedenko, B.V. and Kovalenko, I.N., Vvedenie v teoriyu massovogo obsluzhivaniya (Introduction to Queueing Theory), Moscow: URSS, 2007, 4th ed.

    Google Scholar 

  3. Gnedenko, B.V., Belyaev, Yu.K., and Solov’ev, A.D., Matematicheskie metody v teorii nadezhnosti (Mathematical Methods in Reliability Theory), Moscow: Fizmatgiz, 1966.

    Google Scholar 

  4. Korolyuk, V.S., The Time of Staying Inside a Fixed Set of States for a Semi-Markov Process, Ukr. Mat. Zh., 1965, no. 3, pp. 123–128.

  5. Korolyuk, V.S. and Turbin, A.F., Polumarkovskie protsessy i ikh prilozheniya (Semi-Markov Processes and Their Applications), Kiev: Naukova Dumka, 1999.

    Google Scholar 

  6. Korolyuk, V.S. and Korolyuk, V.V., Stochastic Models of Systems, New York: Kluwer, 1999.

    MATH  Google Scholar 

  7. Boris Vladimirovich Gnedenko v vospominaniyakh uchenikov i soratnikov (Boris Vladimirovich Gnedenko in Recollections of Pupils and Colleagues), collected by Gnedenko, D.B., Gnedenko, B.D., and Gnedenko, E.D., Gnedenko, D.B., Ed., Moscow: KomKniga, 2006.

    Google Scholar 

  8. Mar’yanovich, T.P., Generalization of Erlang Formulas for the Case When Devices May Break and Be Repaired, Ukr. Mat. Zh., 1960, vol. 12, no. 3, pp. 279–286.

    Article  Google Scholar 

  9. Mar’yanovich, T.P., Reliability of Systems with Implicit Reserves, Dopovidi Akad. Nauk URSP, 1961, no. 7, pp. 836–839.

  10. Mar’yanovich, T.P., Reliability of Systems with Mixed Reserves, Dopovidi Akad. Nauk URSP, 1961, no. 8, pp. 964–997.

  11. Mar’yanovich, T.P., Servicing with Regard to Device Failures, in Tr. VI Vsesoyuz. sov. po teorii veroyatnostei i mat. statistike (Proc. of VI All-Union Congress on Probability Theory and Mathematical Statistics), Vil’nyus, 1962, pp. 363–364.

  12. Mar’yanovich, T.P., Some Reliability Problems in Reserved Systems, Ukr. Mat. Zh., 1963, no. 2, pp. 213–219.

  13. Nasirova, T.I., On a Generalization of the Erlang Problem, Tr. Vychisl. Tsentra Akad. Nauk AzSSP, 1963, vol. 2, pp. 3–13.

    Google Scholar 

  14. Yaroshenko, V.N., On a Problem of Queueing Theory, Dopovidi Akad. Nauk URSP, 1964, no. 2, pp. 153–156.

  15. Franken, P., Kenig, D., Arndt, U., and Shmidt, F., Ocheredi i tochechnye protsessy (Queues and Point Processes), Kiev: Naukova Dumka, 1984.

    MATH  Google Scholar 

  16. Ivnitskii, V.A., On an Invariance Condition for Stationary Probabilities in Queueing Networks, Theor. Prob. Appl., 1982, vol. 27, no. 1, pp. 188–192.

    Article  MathSciNet  Google Scholar 

  17. Guseinov, B.T., Generalization of Kovalenko’s Theorem on the Invariance of State Probabilities of Service Systems with Respect to Service Time Distribution, in Lect. 6th Int. Teletraffic Congr., München, 1970.

  18. Gnedenko, D.B., On a Generalization on Erlang Formulas, Zastos. Matem., 1971, vol. 12, pp. 239–242.

    MATH  MathSciNet  Google Scholar 

  19. Sevast’yanov, B.A., Limit Theorem for Markov Processes and Its Application to Telephone Systems with Rejections, Theor. Prob. Appl., 1957, vol. 2, no. 1, pp. 106–116.

    MATH  Google Scholar 

  20. Kovalenko, I.N., On the Condition of Stationary Distributions’ Independence of the Form of the Service Time Distribution Law, Probl. Peredachi Inf., 1963, no. 11, pp. 147–151.

  21. Kelly, F.P., Networks of Queues, Adv. Appl. Prob., 1976, vol. 8, no. 2, pp. 416–432.

    Article  MATH  Google Scholar 

  22. Kelly, F.P. and Williams, R.J., Stochastic Networks, in The IMA Volumes in Math. App., New York: Springer, 1995, vol. 71.

    Google Scholar 

  23. Serfozo, R., Introduction to Stochastic Networks, New York: Springer, 1999.

    MATH  Google Scholar 

  24. Van Dijk, N.M., Queueing Networks and Product Form: A System Approach, New York: Wiley, 1993.

    Google Scholar 

  25. Walrand, J., An Introduction to Queueing Networks, Englewood Cliffs: Prentice Hall, 1988. Translated under the title Vvedenie v teoriyu setei massovogo obsluzhivaniya, Moscow: Mir, 1993.

    MATH  Google Scholar 

  26. Trivedi, K.S., Probability and Statistics with Reliability, Queueing, and Computer Science Applications, New York: Wiley, 2001, 2nd ed.

    Google Scholar 

  27. Yashkov, S.F., Analiz ocheredei v EVM (Analyzing Queues in a Computer), Moscow: Radio i Svyaz’, 1989.

    Google Scholar 

  28. Ivnitskii, V.A., Teoriya setei massovogo obsluzhivaniya (Queueing Theory), Moscow: Fizmatgiz, 2004.

    Google Scholar 

  29. Chao, X., Pinedo, M., and Miyazawa, M., Queueing Networks: Negative Customers, Signals and Product Form, New York: Wiley, 1999.

    MATH  Google Scholar 

  30. Miyazawa, M., The Derivation of Invariance Relations in Complex Queueing Systems with Stationary Inputs, Adv. Appl. Probab., 1983, vol. 15, pp. 874–885.

    Article  MATH  MathSciNet  Google Scholar 

  31. Basharin, G.P. and Tolmachev, A.L., Queueing Theory and Its Applications to Analyzing Informational and Computational Systems, Itogi Nauki Tekhn. Teor. Veroyatn. Mat. Stat. Teor. Kibern., Moscow: VINITI, 1983, vol. 21, pp. 3–119.

    Google Scholar 

  32. Dobrushin, R.L. and Sukhov, Yu.M., An Asymptotic Study of Star-Shaped Message Commuting Networks with a Large Number of Radial Rays, Probl. Peredachi Inf., 1976, vol. 12, no. 1, pp. 70–94.

    MATH  Google Scholar 

  33. Pechinkin, A.V. and Rykov, V.V., On Decomposition of Closed Networks with Dependent Service Times, Autom. Remote Control, 1999, no. 11, part 1, pp. 1568–1576.

  34. Daduna, H., Sojourn Time Distributions in Non-Product Queueing Networks, in Frontiers in Queueing, Dshalalow, J.H., Ed., Boca Raton: CRC Press, 1977, pp. 197–224.

    Google Scholar 

  35. Malinkovskii, Yu.V., Criterion for Product-Form Representation of the Stationary Distribution of States of an Open Markov Queueing Network with Several Classes of Customers, Autom. Remote Control, 1991, no. 4, part 1, pp. 503–509.

  36. Matalytskii, M.A., Study of Networks of Multichannel Queueing Systems with Customers of Different Types, Autom. Remote Control, 1996, no. 9, part 1, pp. 1284–1293.

  37. Ivnitskii, V.A., A Single-Line System with a Queue and Variable Intensities of Input Flow and Service Speed, Lit. Mat. Sb., 1966, vol. 6, no. 1, pp. 122–128.

    Google Scholar 

  38. Ivnitskii, V.A., On Restoring the Characteristics of a Single-Line System with Processing Time Constraints by Observing Its Output Flow, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1969, no. 3, pp. 60–65.

  39. Ivnitskii, V.A., On Restoring the Characteristics of a GI/M/1 System by Observing Its Output Flow, Theor. Prob. Appl., 1977, vol. 22, no. 1, pp. 188–191.

    Article  Google Scholar 

  40. Gnedenko, B.V., On Unloaded Duplication, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1964, no. 4, pp. 3–12.

  41. Gnedenko, B.V., On Duplication with Repairs, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1964, no. 5, pp. 111–118.

  42. Gnedenko, B.V. and Korolev, V.Y., Random Summation: Limit Theorems and Applications, Boca Raton: CRC Press, 1996.

    MATH  Google Scholar 

  43. Gnedenko, B.V., Pavlov, I.B., and Ushakov, I.A., Statistical Reliability Engineering, New York: Wiley, 1999.

    Book  Google Scholar 

  44. Kovalenko, I.N., Atkinson, J.B., and Mykhalevych, K.V., Three Cases of Light-Traffic Insensitivity of the Loss Probability in a GI/G/m/0 Loss System to the Shape of the Service Time Distribution, Queueing Syst., 2003, vol. 45, pp. 245–271.

    Article  MATH  MathSciNet  Google Scholar 

  45. Kovalenko, I.N. and Atkinson, Dzh., Asymptotic Independence Conditions for the Loss Probability in a Queueing Network GI/G/m/0, Kibern. Sist. Analiz, 2002, no. 6, pp. 64–73.

  46. Kovalenko, I.N., A Non-Regenerative Model of a Redundant Repairable System: Bounds for the Unavailability and Asymptotic Insensitivity to Lifetime Distribution, J. Appl. Math. Stoch. Anal., 1994, vol. 9, no. 1.

  47. Kovalenko, I.N., Ergodic and Light-Traffic Properties of a Complex Repairable System, Math. Method Oper. Res., 1997, vol. 45, pp. 387–409.

    Article  MATH  MathSciNet  Google Scholar 

  48. Baum, D. and Kovalenko, I.N., Averaging Properties of a GI/G/m/0 Loss System, Random Oper. Stoch. Equat., 2004, vol. 12, no. 3, pp. 201–210.

    Article  MathSciNet  Google Scholar 

  49. Solov’ev, A.D., Asymptotic Behaviour of the Moment of First Occurrence of a Rare Event in a Regenerating Process, Tekhn. Kibern., 1971, no. 6, pp. 79–89.

  50. Ovchinnikov, V.N., On the Asymptotic Behaviour of the Moment of the First Requirement Loss for Service Depending on System State, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1976, no. 6, pp. 114–120.

  51. Solov’ev, A.D. and Konstantinidis, D.G., Reliability Estimate for a Complex Repairable System with Unbounded Number of Repair Units, Theor. Prob. Appl., 1992, vol. 37, no. 1, pp. 91–94.

    MATH  Google Scholar 

  52. Solov’ev, A.D., Asymptotic Distribution of a Dublirovannogo Element’s Lifetime, Tekhn. Kibern., 1964, no. 5, pp. 119–121.

  53. Solov’ev, A.D., Reserves with Fast Repairs, Tekhn. Kibern., 1970, no. 1, pp. 56–71.

  54. Gnedenko, D.B. and Solov’ev, A.D., A Single Common Reserve Model with Repairs, Tekhn. Kibern., 1974, no. 6, pp. 113–118.

  55. Gnedenko, D.B. and Solov’ev, A.D., Reliability Estimates for Complex Repairable Systems, Tekhn. Kibern., 1975, no. 3, pp. 121–128.

  56. Solov’ev, A.D. and Zaitsev, V.A., Reserves with Incomplete Repairs, Tekhn. Kibern., 1975, no. 1, pp. 72–76.

  57. Zaitsev, V.A. and Solov’ev, A.D., Reserves in Complex Systems, Tekhn. Kibern., 1975, no. 4, pp. 83–92.

  58. Brysina, I.V. and Solov’ev, A.D., Asymptotic Analysis of Queueing Networks under Low Load, Tekhn. Kibern., 1983, no. 3, pp. 40–47.

  59. Makarichev, A.V., Reliability of Systems with Hot Reserves and Replacement with a Loaded Repair Body, Kibern. Sist. Analiz, 1998, no. 2, pp. 181–185.

  60. Makarichev, A.V., Asymptotic Estimates for the Regeneration Period of Complex Repairable Systems for Different Maintenance Disciplines, Elektron. Modelirovanie, 2003, vol. 25, no. 2, pp. 83–97.

    Google Scholar 

  61. Makarichev, A.V., Asymptotic Analysis of the Time before a Complex Repairable System Exits the Set of Faulty States. I, Elektron. Modelirovanie, 2004, vol. 26, no. 2, pp. 51–61.

    Google Scholar 

  62. Makarichev, A.V., Reliability of Complex Systems with Temporary Reserve and Replacement of Repaired Elements into Systems with Minimal Internal Reserves. II, Elektron. Modelirovanie, 2008, vol. 30, no. 6, pp. 99–117.

    Google Scholar 

  63. Kovalenko, I.N., Estimating the Intensity of the Nonmonotonic Loss Flow in a ≤ λ /G/m System, Ukr. Mat. Zh., 2000, vol. 52, no. 9, pp. 1219–1225.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Glushkov Cybernetics Institute, National Academy of Sciences, Kiev, Ukraine

    I. N. Kovalenko

Authors
  1. I. N. Kovalenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © I.N. Kovalenko, 2010, published in Avtomatika i Telemekhanika, 2010, No. 7, pp. 9–14.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kovalenko, I.N. Studying high reliability systems in the probabilistic school of B. V. Gnedenko. Autom Remote Control 71, 1288–1293 (2010). https://doi.org/10.1134/S0005117910070039

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117910070039

Keywords

Search

Navigation