Advertisement

State Reduction in Analysis of a Tandem Queueing System with Correlated Arrivals

  • Vladimir VishnevskyEmail author
  • Andrey Larionov
  • Olga Semenova
  • Roman Ivanov
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 800)

Abstract

Tandem queueing systems often arise in wireless networks modeling. Queueing models are very suitable for network performance evaluation but the system complexity exponential growth (or state space explosion) could make the analysis barely feasible. The paper presents a comparative study of various methods of a state space reduction for markovian arrival processes (MAP) and phase-type distributions (PH) applied to tandem queueing systems. The applied methods include nonlinear optimization, EM-algorithm and linear minimization. While most of the described algorithms are well-studied, a number of issues arises when applying them to a tandem system of a real wireless network. Particularly, it is shown that while all the algorithms could be applied to tandems with a small number of queues, bigger tandems require additional effort to get the appropriable results. Nevertheless, the results presented show that the departure MAPs reduction may help to solve the state space explosion problem.

Keywords

Queueing systems Random process fitting Markov chain space reduction MAP PH Wireless networks modeling 

Notes

Acknowledgements

This work has been financially supported by the Russian Science Foundation and the Department of Science and Technology (India) via grant 16-49-02021 for the joint research project by the V.A. Trapeznikov Institute of control Sciences and the CMS College Kottayam.

References

  1. 1.
    Bobbio, A., Horvath, A., Telek, M.: Matching three moments with minimal acyclic phase type distributions. Stochast. Models 21, 303–326 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bodrog, L., Heindl, A., Horvath, A., Horvath, G., Telek, M.: Current results and open questions on PH and MAP characterization. Numerical Methods for Structured Markov Chains (2007)Google Scholar
  3. 3.
    Bodrog, L., Heindl, A., Horvath, G., Telek, M.: A markovian canonical form of second-order matrix-exponential processes. Eur. J. Oper. Res. 190, 459–477 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Casale, G., Zhang, E.Z., Smirni, E.: Trace data characterization and fitting for markov modeling. Perform. Eval. 67, 61–79 (2010)CrossRefGoogle Scholar
  5. 5.
    Heyman, D., Lucantoni, D.: Modelling multiple IP traffic streams with rate limits. IEEE ACM Trans. Network. 11, 948–958 (2003)CrossRefGoogle Scholar
  6. 6.
    Horvath, A., Horvath, G., Telek, M.: A joint moments based analysis of networks of MAP/MAP/1 queues. Perform. Eval. 67, 759–778 (2010)CrossRefGoogle Scholar
  7. 7.
    Horvath, G., Buchholz, P., Telek, M.: A map fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation. In: Second International Conference on the Quantitative Evaluation of Systems, pp. 124–133 (2005)Google Scholar
  8. 8.
    Horvath, G., Reinecke, P., Telek, M., Wolter, K.: Heuristic representation optimization for efficient generation of PH-distributed random variates. Ann. Oper. Res. 239, 643–665 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Horváth, G., Telek, M.: A canonical representation of order 3 phase type distributions. In: Wolter, K. (ed.) EPEW 2007. LNCS, vol. 4748, pp. 48–62. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-75211-0_5 CrossRefGoogle Scholar
  10. 10.
    Horvath, G., Buchholz, P., Telek, M.: A map fitting approach with independent approximation of the inter-arrival time distribution and the lag correlation. In: Second International Conference on the Quantitative Evaluation of Systems, pp. 124–133 (2005)Google Scholar
  11. 11.
    Horváth, G., Okamura, H.: A fast EM algorithm for fitting marked markovian arrival processes with a new special structure. In: Balsamo, M.S., Knottenbelt, W.J., Marin, A. (eds.) EPEW 2013. LNCS, vol. 8168, pp. 119–133. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40725-3_10 CrossRefGoogle Scholar
  12. 12.
    Hovarth, G.: Butools: Queueing and traffic modeling related functionality for matlab, mathematica and python (2016). https://github.com/ghorvath78/butools
  13. 13.
    Klemm, A., Lindermann, C., Lohmann, M.: Modelling IP traffic using the batch marcovian arrival process. Perform. Eval. 54, 149–173 (2008)CrossRefGoogle Scholar
  14. 14.
    Larionov, A., Ivanov, R.: Pyqumo: Python queueing modeler (2017). https://github.com/larioandr/pyqumo
  15. 15.
    Neuts, M.: A versatile markovian point process. J. Appl. Probab. 16, 764–779 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Okamura, H., Dohi, T.: PH fitting algorithm and its application to reliability engineering. J. Oper. Res. Soc. Japan 59, 72–109 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Okamura, H., Dohi, T.: Faster maximum likelihood estimation algorithms for markovian arrival processes. In: IEEE Sixth International Conference on the Quantitative Evaluation of Systems (QEST 2009) (2009)Google Scholar
  18. 18.
    Telek, M., Horvath, G.: A minimal representation of markov arrival processes and a moments matching method. Perform. Eval. 64, 1153–1168 (2007)CrossRefGoogle Scholar
  19. 19.
    Thummler, A., Buchholz, P., Telek, M.: A novel approach for fitting probability distributions to real trace data with the EM algorithm. In: International Conference on Dependable Systems and Networks (2005)Google Scholar
  20. 20.
    Vishnevski, V., Larionov, A., Ivanov, R.: An open queueing network with a correlated input arrival process for broadband wireless network performance evaluation. In: Dudin, A., Gortsev, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2016. CCIS, vol. 638, pp. 354–365. Springer, Cham (2016). doi: 10.1007/978-3-319-44615-8_31 CrossRefGoogle Scholar
  21. 21.
    Klimenok, V., Dudin, A., Vishnevsky, V.: Tandem queueing system with correlated input and cross-traffic. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2013. CCIS, vol. 370, pp. 416–425. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38865-1_42 CrossRefGoogle Scholar
  22. 22.
    Vishnevsky, V., Dudin, A., Kozyrev, D., Larionov, A.: Methods of performance evaluation of broadband wireless networks along the long transport routes. In: Vishnevsky, V., Kozyrev, D. (eds.) DCCN 2015. CCIS, vol. 601, pp. 72–85. Springer, Cham (2016). doi: 10.1007/978-3-319-30843-2_8 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Vladimir Vishnevsky
    • 1
    Email author
  • Andrey Larionov
    • 1
  • Olga Semenova
    • 1
  • Roman Ivanov
    • 1
  1. 1.V.A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia

Personalised recommendations