Skip to main content
SpringerLink
Log in

Consecutive customer losses in regular and oscillating M X/G/1/n systems

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We derive fast recursions to compute the probability that k or more consecutive customer losses take place during a busy period of a queue, the so called k-CCL probability, for regular and oscillating M X/G/1/n systems.

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. Abramov, V.M.: On a property of a refusals stream. J. Appl. Probab. 34(3), 800–805 (1997)

    Article  Google Scholar 

  2. Abramov, V.M.: On losses in M X/GI/1/n queues. J. Appl. Probab. 38(4), 1079–1080 (2001)

    Article  Google Scholar 

  3. Abramov, V.M.: Stochastic inequalities for single-server loss queueing systems. Stoch. Anal. Appl. 24, 1205–1221 (2006)

    Article  Google Scholar 

  4. Bahary, E., Kolesar, P.: Multilevel bulk service queues. Oper. Res. 20, 406–420 (1972)

    Google Scholar 

  5. Bratiychuk, M., Chydzinski, A.: On the ergodic distribution of oscillating queueing systems. J. Appl. Math. Stoch. Anal. 16(4), 311–326 (2003)

    Article  Google Scholar 

  6. Choi, B.D., Choi, D.I.: Queueing system with queue length dependent service times and its application to cell discarding scheme in ATM networks. IEEE Proc. Commun. 143(1), 5–11 (1996)

    Article  Google Scholar 

  7. Choi, D.I., Knessl, C., Tier, C.: A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks. J. Appl. Math. Stoch. Anal. 12(1), 35–62 (1999)

    Article  Google Scholar 

  8. Chydzinski, A.: The M/G-G/1 oscillating queueing system. Queueing Syst. 42(3), 255–268 (2002)

    Article  Google Scholar 

  9. Chydzinski, A.: On the remaining service time upon reaching a given level in M/G/1 queues. Queueing Syst. 47(1–2), 71–80 (2004)

    Article  Google Scholar 

  10. Chydzinski, A.: The oscillating queue with finite buffer. Perform. Eval. 57(3), 341–355 (2004)

    Article  Google Scholar 

  11. Chydzinski, A.: On the distribution of consecutive losses in a finite capacity queue. WSEAS Trans. Circuits Syst. 4(3), 117–124 (2005)

    Google Scholar 

  12. Cidon, I., Khamisy, A., Sidi, M.: Analysis of packet loss processes in high-speed networks. IEEE Trans. Inform. Theor. 39(1), 98–108 (1993)

    Article  Google Scholar 

  13. de Boer, P.-T.: Analysis and efficient simulation of queueing models of telecommunication systems. PhD thesis, Centre for Telematics and Information Technology, University of Twente, The Netherlands (2000)

  14. de Boer, P.-T., Nicola, V.F.: Hybrid importance sampling estimation of consecutive cell loss probability. AEÜ Int. J. Electron. Commun. 52(3), 133–140 (1998)

    Google Scholar 

  15. de Boer, P.-T., Nicola, V.F., van Ommeren, J.C.W.: The remaining service time upon reaching a high level in M/G/1 queues. Queueing Syst. 39(1), 55–78 (2001)

    Article  Google Scholar 

  16. Fakinos, D., Economou, A.: A new approach for the study of the M X/G/1 queue using renewal arguments. Stoch. Anal. Appl. 19, 151–156 (2001)

    Article  Google Scholar 

  17. Federgruen, A., Tijms, H.C.: Computation of the stationary distribution of the queue size in an M/G/1 queueing system with variable service rate. J. Appl. Probab. 17(2), 515–522 (1980)

    Article  Google Scholar 

  18. Harris, T.J.: The remaining busy period of a finite queue. Oper. Res. 19, 219–223 (1971)

    Google Scholar 

  19. Kant, L., Sanders, W.H.: Loss process analysis of the knockout switch using stochastic activity networks. In Procs. 4th International Conference on Computer Communications and Networks, Sept. 20–23, 1995, pp. 344–349 (1995)

  20. Kwiatkowska, M., Norman, G., Pacheco, A.: Model checking CSL until formulae with random time bounds. In: Lecture Notes in Computer Science, vol. 2399, pp. 152–168. Springer, Berlin (2002)

    Google Scholar 

  21. Kwiatkowska, M., Norman, G., Pacheco, A.: Model checking expected time and expected reward formulae with random time bounds. Comput. Math. Appl. 51(2), 305–316 (2006)

    Article  Google Scholar 

  22. Lee, C.W., Andersland, M.S.: Minimizing consecutive packet loss in real-time ATM sessions. In Procs. of Globecom’94, vol. 2, pp. 935–940 (1994)

  23. Lee, C.W., Andersland, M.S.: Consecutive cell loss controls for leaky-bucket admission systems. In Procs. of Globecom’96, vol. 3, pp. 1732–1738 (1996)

  24. Loris-Teghem, J.: Hysteretic control of an M/G/1 queueing system with two service time distributions and removable server. In: Point Processes and Queuing Problems. Colloq. Math. Soc. János Bolyai, vol. 24, pp. 291–305. North-Holland, Amsterdam (1981)

    Google Scholar 

  25. Miller, L.W.: A note on the busy period of an M/G/1 finite queue. Oper. Res. 23(6), 1179–1182 (1975)

    Google Scholar 

  26. Pacheco, A., Ribeiro, H.: Algorithms for computing moments of the length of busy periods of single-server systems. WSEAS Trans. Comput. 5(11), 2856–2861 (2006)

    Google Scholar 

  27. Pacheco, A., Ribeiro, H.: Consecutive customer loss probabilities in M/G/1/n and GI/M(m)//n systems. In Procs. Workshop on Tools for Solving Structured Markov Chains, Pisa, Italy, October 10, 2006

  28. Pacheco, A., Ribeiro, H.: Moments of the duration of busy periods of M X/G/1/n systems. Probab. Eng. Inf. Sc. (2008, in print)

  29. Peköz, E.A.: On the number of refusals in a busy period. Probab. Eng. Inf. Sci. 13(1), 71–74 (1999)

    Article  Google Scholar 

  30. Peköz, E.A., Righter, R., Xia, C.H.: Characterizing losses during busy periods in finite buffer systems. J. Appl. Probab. 40(1), 242–249 (2003)

    Google Scholar 

  31. Ribeiro, H.: Customer loss probabilities and other performance measures of regular and oscillating systems. PhD thesis, Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Portugal (2007)

  32. Righter, R.: A note on losses in M/GI/1/n queues. J. Appl. Probab. 36(4), 1240–1243 (1999)

    Article  Google Scholar 

  33. Sriram, K., McKinney, R.S., Sherif, M.H.: Voice packetization and compression in broadband ATM networks. IEEE J. Select. Areas Commun. 9(3), 294–304 (1991)

    Article  Google Scholar 

  34. Takine, T., Suda, T., Hasegawa, T.: Cell loss and output process analyses of a finite-buffer discrete-time ATM queueing system with correlated arrivals. IEEE Trans. Commun. 43(2-4), 1022–1037 (1995)

    Article  Google Scholar 

  35. Vijaya Laxmi, P.V., Gupta, U.C.: Analysis of finite-buffer multi-server queues with group arrivals: GI X/M/c/N. Queueing Syst. 36(1–3), 125–140 (2000)

    Article  Google Scholar 

  36. Willmot, G.E.: On recursive evaluation of mixed-Poisson probabilities and related quantities. Scand. Actuar. J. 2, 114–133 (1993)

    Google Scholar 

  37. Wolff, R.W.: Losses per cycle in a single-server queue. J. Appl. Probab. 39(4), 905–909 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and CEMAT, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    António Pacheco

  2. Department of Mathematics and CEMAT, Escola Superior de Tecnologia e Gestão, Polytechnic Institute of Leiria, Campus 2, Morro do Lena–Alto do Vieiro, 2411-901, Leiria, Portugal

    Helena Ribeiro

Authors
  1. António Pacheco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helena Ribeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to António Pacheco.

Additional information

This research was supported in part by Programa de Formação Avançada de Docentes do Ensino Superior Medida 5/Acção 5.3 (PRODEP III) and the Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI) of the Fundação para a Ciência e a Tecnologia (FCT), cofinanced by the European Community fund FEDER, and the projects POSC/EIA/60061/2004, Euro-NGI, and Euro-FGI.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pacheco, A., Ribeiro, H. Consecutive customer losses in regular and oscillating M X/G/1/n systems. Queueing Syst 58, 121–136 (2008). https://doi.org/10.1007/s11134-008-9063-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-008-9063-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation