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On minimal representations of Rational Arrival Processes

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  • Peter Buchholz
  • Miklós Telek

Abstract

Rational Arrival Processes (RAPs) form a general class of stochastic processes which include Markovian Arrival Processes (MAPs) as a subclass. In this paper we study RAPs and their representations of different sizes. We show some transformation methods between different representations and present conditions to evaluate the size of the minimal representation. By using some analogous results from linear systems theory, a minimization approach is defined which allows one to transform a RAP (from a redundant high dimension) into one of its minimal representations. An algorithm for computing a minimal representation is also given. Furthermore, we extend the approach to RAPs with batch arrivals (BRAPs) and to RAPs with arrivals of different customer types (MRAPs). Copyright Springer Science+Business Media, LLC 2013

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  • Peter Buchholz & Miklós Telek, 2013. "On minimal representations of Rational Arrival Processes," Annals of Operations Research, Springer, vol. 202(1), pages 35-58, January.
    • Handle: RePEc:spr:annopr:v:202:y:2013:i:1:p:35-58:10.1007/s10479-011-1001-5
    DOI: 10.1007/s10479-011-1001-5
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    References listed on IDEAS

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    1. Levente Bodrog & András Horváth & Miklós Telek, 2008. "Moment characterization of matrix exponential and Markovian arrival processes," Annals of Operations Research, Springer, vol. 160(1), pages 51-68, April.
    2. Asmussen, Søren & Bladt, Mogens, 1999. "Point processes with finite-dimensional conditional probabilities," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 127-142, July.
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    Cited by:

    1. Casale, Giuliano & Sansottera, Andrea & Cremonesi, Paolo, 2016. "Compact Markov-modulated models for multiclass trace fitting," European Journal of Operational Research, Elsevier, vol. 255(3), pages 822-833.
    2. Miklós Telek, 2022. "The two-matrix problem," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 265-267, April.
    3. Gábor Horváth & Philipp Reinecke & Miklós Telek & Katinka Wolter, 2016. "Heuristic representation optimization for efficient generation of PH-distributed random variates," Annals of Operations Research, Springer, vol. 239(2), pages 643-665, April.
    4. Peter Buchholz & Jan Kriege, 2017. "Fitting correlated arrival and service times and related queueing performance," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 337-359, April.

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