Skip to main content
SpringerLink
Log in

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

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

A GI/G/m/0 loss system is considered. Three cases of light-traffic insensitivity of the loss probability to the shape of the service time distribution, given its first moment, are investigated in a triangle array setting.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Anastasi and L. Lenzini, QoS provided by the IEEE 802.11 wireless LAN to advanced data applications: A simulation analysis, ACM Wireless Networks 6(2) (2000) 99–108.

    Google Scholar 

  2. V.V. Anisimov, Asymptotic analysis of switching queueing systems in conditions of low and heavy loading, in: Matrix–Analytic Methods for Stochastic Models, Lecture Notes in Pure and AppliedMathematics, Vol. 183 (Marcel Dekker, New York, 1996) 241–260.

    Google Scholar 

  3. V.V. Anisimov, Asymptotic analysis of stochastic models of hierarchic structure and applications in queueing models, in: Advances in Matrix Analytic Methods for Stochastic Models, eds. A.S. Alfa and S.R. Chakravarthy (Notable Publ., USA, 1998) pp. 237–259.

    Google Scholar 

  4. V.V. Anisimov and M. Kurtulus, Markov retrial queueing models in light traffic conditions, Cybernet. Systems Anal. 36(6) (2001).

  5. V.V. Anisimov and J. Sztrik, Asymptotic analysis of some controlled finite–source queueing systems, Acta Cybernet. 9 (1989) 27–39.

    Google Scholar 

  6. V.V. Anisimov and J. Sztrik, Asymptotic analysis of some complex renewable systems operating in random environments, European J. Oper. Res. 41 (1989) 162–168.

    Google Scholar 

  7. V.V. Anisimov and J. Sztrik, Reliability analysis of complex renewable system with fast repair, J. Inform. Process. Cybernet. 25 (11/12) (1989) 573–580.

    Google Scholar 

  8. V.V. Anisimov, O.K. Zakusilo and V.S. Donchenko, The Elements of Queueing Theory and Asymptotic Analysis of Systems (Vyshcha Shkola at Kyiv University, Kyiv, 1987) (in Russian).

    Google Scholar 

  9. S. Asmussen, Applied Probability and Queues (Wiley, New York, 1987).

    Google Scholar 

  10. S. Asmussen, Light traffic equivalence in single–server queues, Ann. Appl. Probab. 2 (1992) 555–574.

    Google Scholar 

  11. J.B. Atkinson and I.N. Kovalenko, On the practical insensitivity of the availability of some redundant repairable systems to the lifetime distribution in light traffic, in: Proc. of the Internat. Conf. on Probability Analysis of Rare Events, Aviation University, Riga, 1999, pp. 83–91.

    Google Scholar 

  12. F. Baccelli and V. Schmidt, Taylor–series expansions for Poisson–driven (max. +)–linear systems, Ann. Appl. Probab. 6 (1) (1996) 138–185.

    Google Scholar 

  13. B. Blaszczyszyn, A. Frey and V. Schmidt, Light–traffic approximations for Markov–modulated multiserver queues, INTERNET (2002) 1–23.

  14. B. Blaszczyszyn, T. Rolski and V. Schmidt, Light–traffic approximations in queues and related stochastic models, in: Advances in Queueing, ed. J.H. Dshalalow (CRC Press, Boca Raton, FL, 1995) pp. 379–406.

    Google Scholar 

  15. P. Bloomfield and D.R. Cox, A low traffic approximation for queues, J. Appl. Probab. 9 (1972) 832–840.

    Google Scholar 

  16. J.–L. Bon and P. Pamphile, Reliability evaluation of Markov models from Monte Carlo methods, J. Appl. Probab. 9 (1972) 92–97.

    Google Scholar 

  17. A.A. Borovkov, Asymptotical Methods in Queueing Theory (Nauka, Moscow, 1980) (in Russian).

    Google Scholar 

  18. A.A. Borovkov, Asymptotic expansions for functionals of dilations of point processes, INTERNET (2002) 1–27.

  19. R.J. Boucherie, Product forms on backward traffic equations, J. Appl. Probab. 32 (1995) 519–533.

    Google Scholar 

  20. B.D. Bunday and R.E. Scraton, The G/M/r machine interference problem, European J. Oper. Res. 4 (1980) 399–402.

    Google Scholar 

  21. [21]D.Y. Burmana and D.R. Smith, A light traffic theorem for multi–server queues, Math. Oper. Res. 8 (1983) 15–25.

    Google Scholar 

  22. D.J. Daley and T. Rolski, Light–traffic approximations in many–server queues, Adv. in Appl. Probab. 24 (1992) 202–218.

    Google Scholar 

  23. D.J. Daley and T. Rolski, Light–traffic approximations in general stationary single–server queues, Stochastic Process. Appl. 49 (1994) 141–158.

    Google Scholar 

  24. T. Erhardsson, On the number of lost customers in stationary loss systems in the light traffic case, INTERNET (2002) 1–19.

  25. P. Franken, D. König, U. Arndt and V. Schmidt, Queues and Point Processes (Wiley, New York, 1983).

    Google Scholar 

  26. I.B. Gertsbakh, Asymptotic methods in reliability theory: A review, Adv. in Appl. Probab. 16 (1984) 147–175.

    Google Scholar 

  27. B.V. Gnedenko and I.N. Kovalenko, Introduction to Queueing Theory (Birkhäuser, Boston, 1989).

    Google Scholar 

  28. D.B. Gnedenko and A.D. Soloviev, Estimation of reliability of complex repairable systems, Tekh. Kibernet. 13(3) (1975) 89–96 (in Russian).

    Google Scholar 

  29. V.A. Gryshchenko, A Poissonian approximation for the flow of lost customers in a multi–channel queueing system in the condition of rare losses, Kibernetika 4 (1985) 83–87 (in Russian).

    Google Scholar 

  30. H.L. Hartmann, Near explicit state probabilities of multi–services loss systems, INTERNET (2002) 1–25.

  31. V.V. Kalashnikov, Topics on Regenerative Processes (CRC Press, Boca Raton, FL, 1994).

    Google Scholar 

  32. D.G. Konstantinidis, The principle of a monotone failure path of a complex repairable system, Vestnik Moskow Univ. 3 (1990) 7–13 (in Russian).

    Google Scholar 

  33. V.S. Korolyuk and A.F. Turbin, Markov Renewal Processes and their Application to Systems Reliability (Naukova Dumka, Kyiv, 1982) (in Russian).

    Google Scholar 

  34. I.N. Kovalenko, Rare Event Analysis in the Estimation of Systems Reliability and Performance (Sovetskoe Radio, Moscow, 1980) (in Russian).

    Google Scholar 

  35. I.N. Kovalenko, Rare events in queueing systems, A survey, Queueing Systems 16(1) (1994) 1–49.

    Google Scholar 

  36. I.N. Kovalenko, A nonregenerative model of a redundant repairable system: Bounds for the unavailability and asymptotic insensitivity to lifetime distribution, J. Appl. Math. Stochastic. Anal. 9(1) (1994) 1–49.

    Google Scholar 

  37. I.N. Kovalenko, Approximation of queues via small–parameter method, in: Advances in Queueing, ed. J.H. Dshalalow (CRC Press, Boca Raton, FL, 1995) pp. 481–506.

    Google Scholar 

  38. I.N. Kovalenko, Ergodic and light–traffic properties of a complex repairable system, Math. Methods Oper. Res. 45 (1997) 387–409.

    Google Scholar 

  39. I.N. Kovalenko, A bound for the intensity of the flow of non–monotone failures in a (? ?)/ G / mgueueing system, Ukrainian Mat. J. 52(9) (2000) 1219–1225 (in Russian; translated).

    Google Scholar 

  40. I.N. Kovalenko and J.B. Atkinson, Conditions for the light traffic insensitivity of the loss probability in a GI / G/ m /0 queueing system, Cybernet Systems Anal. 6 (2002) 64–73 (in Russian, to be translated).

    Google Scholar 

  41. I.N. Kovalenko, N. Yu. Kuznetsov and P.A. Pegg, Mathematical Theory of Reliability of Time–Dependent Systems with Practical Applications (Wiley, Chichester, 1997).

    Google Scholar 

  42. M. Miyazawa, The derivation of invariance relations in complex queueing systems with stationary inputs, Adv. in Appl. Probab. 15 (1983) 874–885.

    Google Scholar 

  43. [43]M. Nabe and M.M.H. Miyahara, Analysis and modeling of World Wide Web traffic for capacity dimensioning in Internet access lines, Performance Evaluation 34 (1998) 249–271.

    Google Scholar 

  44. M. Naldi, Measurement–based modelling of Internet dial–up access connections, Computer Networks 31 (1999) 2381–2390.

    Google Scholar 

  45. S.A. Nozaki and S.M. Ross, Approximations in finite–capacity multiserver queues with Poisson arrivals, J. Appl. Probab. 15 (1978) 826–834.

    Google Scholar 

  46. H.B. Pavlov, An inverse problem in queueing theory, SERDIKA B'lgarsko Matem. Spisanie 5 (1979) 287–291 (in Russian).

    Google Scholar 

  47. M.I. Reiman and B. Simon, Open queueing systems in light traffic,Math. Oper. Res. 14 (1989) 26–59.

    Google Scholar 

  48. A.D. Soloviev, The redundancy with a fast repair, Tekh. Kibernet. 8(1) (1970) 49–64 (in Russian).

    Google Scholar 

  49. A.D. Soloviev and N.G. Karaseva, The estimation of the mean lifetime of repairable systems, Vestnik Moskow Univ. 5(1) (1998) 25–29 (in Russian).

    Google Scholar 

  50. H. Tijms, Stochastic Models: An Algorithmic Approach (Wiley, New York, 1994).

    Google Scholar 

  51. H.C. Tijms, M.H. Van Hoorn and A. Federgruen, Approximations for the steady–state probabilities in the M / G /c queue, Adv. in Appl. Probab. 13 (1981) 186–206.

    Google Scholar 

  52. N. Van Dijk and H.C. Tijms, Insensitivity in two–node blocking networks models with applications I, in: Teletraffic Analysis and Computer Performance Evaluation, eds. O.J. Boxma, J.W. Cohen and H.C. Tijms (North Holland, Amsterdam, 1986).

    Google Scholar 

  53. C.–L. Wang, Light–traffic approximations for regenerative queueing processes, Adv. in Appl. Probab. 29 (1997) 1060–1080.

    Google Scholar 

  54. P. Whittle, Systems in Stochastic Equilibrium (Wiley, Chichester /New York, 1986).

    Google Scholar 

  55. M.A. Zazanis, Analyticity of Poisson–driven stochastic systems, Adv. in Appl. Probab. 24 (1992) 532–541.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, 40 Glushkov Avenue, 03680, MSP, Kyiv 187, Ukraine

    I.N. Kovalenko & K.V. Mykhalevych

  2. STORM Research Centre, London Metropolitan University, 166-200 Holloway Road, London, N7 8DB, UK

    J.B. Atkinson & K.V. Mykhalevych

Authors
  1. I.N. Kovalenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J.B. Atkinson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K.V. Mykhalevych
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kovalenko, I., Atkinson, J. & Mykhalevych, K. 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 Systems 45, 245–271 (2003). https://doi.org/10.1023/A:1027328719269

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1027328719269

Search

Navigation