Skip to main content
Log in

Transient analysis of the MMPP/G/1/K queue

  • Published:
Telecommunication Systems Aims and scope Submit manuscript

Abstract

In this paper the transient behaviour of a finite-buffer queue fed by the Markov-modulated Poisson process is studied. The results include formulas for the transforms of transient queue size distribution, transient full buffer probability and transient delay (workload). Computational issues are discussed and numerical samples presented.

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

Access this article

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. W. Leland, M. Taqqu, W. Willinger and D. Wilson, On the self-similar nature of ethernet traffic (extended version), IEEE/ACM Transactions on Networking 2(1) (1994) 1–15.

    Google Scholar 

  2. A. Andersen and B. Nielsen, A Markovian approach for modeling packet traffic with long-range dependence, IEEE Journal on Selected Areas in Communications 16(5) (1998) 719–732.

    Article  Google Scholar 

  3. L. Deng and J. Mark, Parameter estimation for Markov modulated Poisson processes via the EM algorithm with time discretization, Telecommunication Systems 1 (1993) 321–338.

    Article  Google Scholar 

  4. S. Li and C. Hwang, On the convergence of traffic measurement and queuing analysis: A statistical match and queuing (SMAQ) tool, IEEE/ACM Transactions on Networking (Feb. 1997) 95–110.

  5. S. Kang and D. Sung, Two-state MMPP modelling of ATM superposed traffic streams based on the characterisation of correlated interarrival times, in: Proc. of IEEE GLOBECOM ’95, pp. 1422–1426.

  6. S. Robert and J. Le Boudec, New models for self-similar traffic, Performance Evaluation 30(1/2) (1997).

  7. T. Yoshihara, S. Kasahara and Y. Takahashi, Practical time-scale fitting of self-similar traffic with Markov-modulated Poisson process, Telecommunication Systems 17(1/2) (2001) 185–211.

    Article  Google Scholar 

  8. A. Adas, Traffic models in broadband networks, IEEE Communications Magazine (7) (1997) 82–89.

  9. H. Heffes and D. Lucantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. Select. Areas Commun. 4(6) (1986) 856–868.

    Article  Google Scholar 

  10. S. Shah-Heydari and T. Le-Ngoc, MMPP models for multimedia traffic, Telecommunication Systems 15(3–4) (2000) 273–293.

    Article  Google Scholar 

  11. T.C. Wong, J.W. Mark and K.C. Chua, Delay performance of voice and MMPP video traffic in a cellular wireless ATM network, IEE Proceedings Communications 148 (2001) 302–309.

    Google Scholar 

  12. G.L. Wu and J.W. Mark, Computational methods for performance evaluation of a statistical multiplexer supporting bursty traffic, IEEE/ACM Transactions on Networking 4(3) (1996) 386–397.

    Google Scholar 

  13. Y.H. Kim and C.K. Un, Performance analysis of statistical multiplexing for heterogeneous bursty traffic in ATM network, IEEE Trans. Commun. 42(2–4) (1994) 745–753.

    Article  Google Scholar 

  14. P. Skelly, M. Schwartz and S. Dixit, A histogram-based model for video traffic behavior in an ATM multiplexer, IEEE/ACM Trans. Netw. 1(4) (1993) 446–459.

    Article  Google Scholar 

  15. T. Ryden, An EM algorithm for parameter estimation in Markov modulated Poisson processes, Comput. Stat. Data Anal. 21 (1996) 431–447.

    Article  Google Scholar 

  16. H. Ge, U. Harder and P. G. Harrison, Parameter estimation for MMPPs using the EM algorithm, Proc. UKPEW, July (2003).

  17. P. Salvador, R. Valadas and A. Pacheco, Multiscale fitting procedure using markov modulated poisson processes, Telecommunication Systems 23(1–2) (2003) 123–148.

    Article  Google Scholar 

  18. A. Klemm, C. Lindemann and M. Lohmann, Modeling IP traffic using the batch markovian arrival process, Performance Evaluation 54(2) (2003).

  19. W. Fischer and K. Meier-Hellstern, The Markov-modulated Poisson process (MMPP) cookbook, Performance Evaluation 18(2) (1992) 149–171.

    Article  Google Scholar 

  20. D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Commun. Stat., Stochastic Models 7(1) (1991) 1–46.

    Article  Google Scholar 

  21. A. Baiocchi and N. Blefari-Melazzi, Steady-state analysis of the MMPP/G/1/K queue, IEEE Trans. Commun. 41(4) (1992) 531–534.

    Article  Google Scholar 

  22. D.M. Lucantoni, G.L. Choudhury and W. Whitt, The transient BMAP/G/1 queue, Commun. Stat., Stochastic Models 10(1) (1994) 145–182.

    Article  Google Scholar 

  23. D. Lucantoni, Further transient analysis of the BMAP/G/1 queue, Commun. Stat., Stochastic Models 14(1–2) (1998) 461–478.

    Article  Google Scholar 

  24. L.-M. Le Ny and B. Sericola, Transient analysis of the BMAP/PH/1 queue, International Journal of Simulation: Systems, Science & Technology. Special Issue on Analytical & Stochastic Modeling Techniques 3(3–4) (2002).

  25. B. Van Houdt and C. Blondia, QBDs with marked time epochs: a framework for transient performance measures, Proc. of QEST 2005, Torino, Italy (2005) pp. 210–219.

  26. L. Kulkarni and S.-Q. Li, Transient behaviour of queueing systems with correlated traffic, Perform. Eval. 27 and 28 (1996) 117–145.

    Google Scholar 

  27. D.-S. Lee and S.-Q. Li, Transient analysis of multi-server queues with Markov-modulated Poisson arrivals and overload control, Perform. Eval. 16(1–3) (1992) 49–66.

    Article  Google Scholar 

  28. S. Ross, Approximating transition probabilities and mean occupation times in continuous-time Markov chains, Probability in the Engineering and Informational Sciences 1 (1987) 251–264.

  29. R.M.L.R. Carmo, E. de Souza e Silva and R. Marie, Efficient solutions for an approximation technique for the transient analysis of Markovian models, Technical report, IRISA Publication Interne. N 1067 (1996).

  30. B. Van Houdt and C. Blondia, Approximated transient queue length and waiting time distributions via steady state analysis, Stochastic Models 21(2/3) (2005) 725–744.

    Article  Google Scholar 

  31. W.S. Korolyuk, Boundary Problems for Compound Poisson Processes (in Russian), Naukowa Dumka, Kiev (1975).

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

    Article  Google Scholar 

  33. A. Chydzinski, The oscillating queue with finite buffer, Performance Evaluation 57(3) (2004) 341–355.

    Article  Google Scholar 

  34. J. Abate, G.L. Choudhury and W. Whitt, An introduction to numerical transform inversion and its application to probability models, in: W. Grassman (Ed.), Chapter in Computational Probability, Kluwer, Boston (2000) pp. 257–323.

    Google Scholar 

  35. H. Takagi, Queueing Analysis. Vol. 2. Finite Systems, North-Holland, Amsterdam (1993).

    Google Scholar 

  36. A. Pacheco and N.U. Prabhu, Markov-additive processes of arrivals, in: J.H. Dshalalow (Ed.), Advances in Queueing: Theory, Methods and Open Problems, Chap. 6, CRC Press, Boca Raton, Florida (1995) pp. 167–194.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrzej Chydzinski.

Additional information

This material is based upon work supported by the Polish Ministry of Scientific Research and Information Technology under Grant No. 3 T11C 014 26.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chydzinski, A. Transient analysis of the MMPP/G/1/K queue. Telecommun Syst 32, 247–262 (2006). https://doi.org/10.1007/s11235-006-9001-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11235-006-9001-5

Keywords

Navigation