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Canonical Form of Order-2 Non-stationary Markov Arrival Processes

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Computer Performance Engineering (EPEW 2015)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9272))

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Abstract

Canonical forms of Markovian distributions and processes provide an efficient way of describing these structures by eliminating the redundancy of the general description. Canonical forms of order-2 stationary Markov arrival processes (MAPs) have already been established for both continuous and discrete time. In this paper we prove that the canonical form of continuous time MAPs can be naturally extended to their non-stationary generalisations. We also prove that the equivalence proven for order-2 stationary Markov arrival processes and rational arrival processes also holds for the non-stationary counterparts.

This work is partially supported by the OTKA K101150 projects.

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References

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

    Article  Google Scholar 

  2. Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics 419–441 (1996)

    Google Scholar 

  3. Bladt, M.: A review on phase-type distributions and their use in risk theory. ASTIN Bulletin 35, 145–161 (2005). http://journals.cambridge.org/article_S0515036100014100

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodrog, L., Heindl, A., Horváth, G., Telek, M.: A Markovian canonical form of second-order matrix-exponential processes. European Journal of Operation Research 190, 459–477 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buchholz, P., Horvath, A., Telek, M.: Stochastic Petri nets with low variation matrix exponentially distributed firing time. International Journal of Performability Engineering 7, 441–454 (2011). special issue on Performance and Dependability Modeling of Dynamic Systems

    Google Scholar 

  6. Buchholz, P., Panchenko, A.: A two-step EM algorithm for MAP fitting. In: Aykanat, C., Dayar, T., Körpeoğlu, İ. (eds.) ISCIS 2004. LNCS, vol. 3280, pp. 217–227. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  7. Buchholz, P., Telek, M.: On minimal representation of rational arrival processes. Annals of Operations Research 202, 35–58 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fackrell, M.: Modelling healthcare systems with phase-type distributions. Health Care Management Science 12(1), 11–26 (2009). http://dx.doi.org/10.1007/s10729-008-9070-y

    Article  MathSciNet  Google Scholar 

  9. Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. In: Proceedings IEEE INFOCOM 1997. Sixteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Driving the Information Revolution, vol. 3, pp. 1096–1104. IEEE (1997)

    Google Scholar 

  10. Fischer, W., Meier-Hellstern, K.: The Markov-modulated Poisson process (MMPP) cookbook. Performance evaluation 18(2), 149–171 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Horváth, A., Telek, M.: A Markovian point process exhibiting multifractal behaviour and its application to traffic modeling. In: Proceedings of 4th International Conference on Matrix-Analytic Methods in Stochastic models, pp. 183–208 (2002)

    Google Scholar 

  12. Horváth, G.A., 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)

    Chapter  Google Scholar 

  13. Houdt, B.V., Alfa, A.S.: Response time in a tandem queue with blocking, Markovian arrivals and phase-type services. Operations Research Letters 33(4), 373–381 (2005). http://www.sciencedirect.com/science/article/pii/S0167637704001245

    Article  MathSciNet  MATH  Google Scholar 

  14. Khayari, R.E.A., Sadre, R., Haverkort, B.R.: Fitting world-wide web request traces with the EM-algorithm. Performance Evaluation 52(2), 175–191 (2003)

    Article  Google Scholar 

  15. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics (1999). https://books.google.com/books?id=J31a3P69K8IC

  16. Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Communications in Statistics. Stochastic Models 7(1), 1–46 (1991). http://dx.doi.org/10.1080/15326349108807174

    Article  MathSciNet  MATH  Google Scholar 

  17. Mészáros, A., Telek, M.: Canonical representation of discrete order 2 MAP and RAP. In: Balsamo, M.S., Knottenbelt, W.J., Marin, A. (eds.) EPEW 2013. LNCS, vol. 8168, pp. 89–103. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  18. Mitchell, K., van de Liefvoort, A.: Approximation models of feed-forward G/G/1/N queueing networks with correlated arrivals. Performance Evaluation 51(2), 137–152 (2003)

    Article  Google Scholar 

  19. Neuts, M.F.: Models based on the Markovian arrival process. IEICE Transactions on Communications 75(12), 1255–1265 (1992)

    Google Scholar 

  20. Papp, J., Telek, M.: Canonical representation of discrete phase type distributions of order-2 and 3. In: Proc. of UK Performance Evaluation Workshop, UKPEW 2013 (2013)

    Google Scholar 

  21. Puhalskii, A.A., Reiman, M.I.: The multiclass GI/PH/N queue in the Halfin-Whitt regime. Advances in Applied Probability 32(2), 564–595 (2000). http://www.jstor.org/stable/1428204

    Article  MathSciNet  MATH  Google Scholar 

  22. Rodriguez, J., Lillo, R.E., Ramirez-Cobo, P.: A canonical form for the non-stationary two-state Markovian arrival process (2015). submitted for publication

    Google Scholar 

  23. Rydn, T.: An EM algorithm for estimation in Markov-modulated Poisson processes. Computational Statistics & Data Analysis 21(4), 431–447 (1996). http://www.sciencedirect.com/science/article/pii/0167947395000259

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Budapest University of Technology and Economics, Magyar Tudósok krt. 2, Budapest, 1117, Hungary

    András Mészáros & Miklós Telek

  2. MTA-BME Information Systems Research Group, Magyar Tudósok krt. 2, Budapest, 1117, Hungary

    András Mészáros & Miklós Telek

Authors
  1. András Mészáros
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  2. Miklós Telek
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Corresponding author

Correspondence to András Mészáros .

Editor information

Editors and Affiliations

  1. Universidad Rey Juan Carlos, Mostoles (Madrid), Spain

    Marta Beltrán

  2. Imperial College London, London, United Kingdom

    William Knottenbelt

  3. Imperial College London, London, United Kingdom

    Jeremy Bradley

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Mészáros, A., Telek, M. (2015). Canonical Form of Order-2 Non-stationary Markov Arrival Processes. In: Beltrán, M., Knottenbelt, W., Bradley, J. (eds) Computer Performance Engineering. EPEW 2015. Lecture Notes in Computer Science(), vol 9272. Springer, Cham. https://doi.org/10.1007/978-3-319-23267-6_11

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  • DOI: https://doi.org/10.1007/978-3-319-23267-6_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23266-9

  • Online ISBN: 978-3-319-23267-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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