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Moment characterization of matrix exponential and Markovian arrival processes

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Abstract

This paper provides a general framework for establishing the relation between various moments of matrix exponential and Markovian processes. Based on this framework we present an algorithm to compute any finite dimensional moments of these processes based on a set of required (low order) moments. This algorithm does not require the computation of any representation of the given process. We present a series of related results and numerical examples to demonstrate the potential use of the obtained moment relations.

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References

  • Asmussen, S., & Bladt, M. (1996). Renewal theory and queueing algorithms for matrix-exponential distributions. In Proc. 1st int. conference on matrix-analytic methods in stochastic models (pp. 313–341).

  • Asmussen, S., & Bladt, M. (1999). Point processes with finite-dimensional conditional probabilities. Stochastic Processes and their Applications, 82, 127–142 (16), July.

    Article  Google Scholar 

  • Asmussen, S., & O’Cinneide, C. A. (1997). Matrix-exponential distributions—distributions with a rational Laplace transform. In S. Kotz & C. Read (Eds.), Encyclopedia of statistical sciences (pp. 435–440). New York: Wiley.

    Google Scholar 

  • Bladt, M., & Neuts, M. (2003). Matrix-exponential distributions: calculus and interpretations via flows. Stochastic Models, 19(1), 113–124.

    Article  Google Scholar 

  • Bodrog, L., Heindl, A., Horvath, G., & Telek, M. (2007). A Markovian canonical form of second-order matrix-exponential processes. European Journal of Operational Research, in press.

  • Fackrell, M. (2005). Fitting with matrix-exponential distributions. Stochastic Models, 21, 377–400.

    Article  Google Scholar 

  • Gragg, W. B., & Linquist, A. (1983). On the partial realization problem. Linear Algebra and Applications, 50, 277–319.

    Article  Google Scholar 

  • He, Q.-M., & Zhang, H. (2007). On matrix-exponential distributions. Advances in Applied Probability, 39(1), 271–292.

    Article  Google Scholar 

  • Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. Philadelphia: SIAM.

    Google Scholar 

  • Lipsky, L. (1992). Queueing theory: a linear algebraic approach. New York: MacMillan.

    Google Scholar 

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

    Article  Google Scholar 

  • Mitchell, K., Sohraby, K., van de Liefvoort, A., & Place, J. (2000). Approximation models of wireless cellular networks using moment matching. In Proc. conf. on computer communications (IEEE Infocom) (pp. 189–197).

  • Neuts, M. (1979). A versatile Markovian point process. Journal of Applied Probability, 16, 764–779.

    Article  Google Scholar 

  • Neuts, M. (1981). Matrix-geometric solutions in stochastic models. Baltimore: John Hopkins University Press.

    Google Scholar 

  • Telek, M., & Heindl, A. (2002). Matching moments for acyclic discrete and continuous phase-type distributions of second order. International Journal of Simulation Systems, Science & Technology, 3(3–4), 47–57. Special Issue on: Analytical & Stochastic Modelling Techniques.

    Google Scholar 

  • van de Liefvoort, A. (1990). The moment problem for continuous distributions (Technical Report). University of Missouri, WP-CM-1990-02, Kansas City.

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Correspondence to András Horváth.

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This work is partially supported by the Italian-Hungarian bilateral R&D programme, by OTKA grant n. T-34972, by MIUR through PRIN project Famous and by EEC project Crutial.

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Bodrog, L., Horváth, A. & Telek, M. Moment characterization of matrix exponential and Markovian arrival processes. Ann Oper Res 160, 51–68 (2008). https://doi.org/10.1007/s10479-007-0296-8

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  • DOI: https://doi.org/10.1007/s10479-007-0296-8

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