Abstract
The class of order 3 phase type distributions (PH(3)) is known to be a proper subset of the class of order 3 matrix exponential distributions (ME(3)). In this paper we investigate the relation of these two sets for what concerns their moment bounds. To this end we developed a procedure to check if a matrix exponential function of order 3 defines a ME(3) distribution or not. This procedure is based on the time domain analysis of the density function. The proposed procedure requires the numerical solution of a transcendent equation in some cases.
The presented moment bounds are based on some unproved conjectures which are verified only by numerical investigations.
This work is partially supported by the NAPA-WINE FP7 project and by the OTKA K61709 grant.
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References
Asmussen, S., O’Cinneide, C.A.: Matrix-exponential distributions – distributions with a rational Laplace transform. In: Kotz, S., Read, C. (eds.) Encyclopedia of Statistical Sciences, pp. 435–440. John Wiley & Sons, New York (1997)
Bean, N.G., Nielsen, B.F.: Quasi-birth-and-death processes with rational arrival process components. Technical report, Informatics and Mathematical Modelling, Technical University of Denmark, DTU, IMM-Technical Report-2007-20 (2007)
Bobbio, A., Horváth, A., Telek, M.: Matching three moments with minimal acyclic phase type distributions. Stochastic models, 303–326 (2005)
Éltető, T., Rácz, S., Telek, M.: Minimal coefficient of variation of matrix exponential distributions. In: 2nd Madrid Conference on Queueing Theory, Madrid, Spain (July 2006) (abstract)
Fackrell, M.W.: Characterization of matrix-exponential distributions. Technical report, School of Applied Mathematics, The University of Adelaide, Ph. D. thesis (2003)
Horváth, G., Telek, M.: On the canonical representation of phase type distributions. Performance Evaluation (2008), doi:10.1016/j.peva.2008.11.002
Latouche, G., Ramaswami, V.: Introduction to matrix analytic methods in stochastic modeling. SIAM, Philadelphia (1999)
Lipsky, L.: Queueing Theory: A linear algebraic approach. MacMillan, New York (1992)
Neuts, M.: Matrix-Geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore (1981)
Pérez, J.F., Van Velthoven, J., Van Houdt, B.: Q-mam: A tool for solving infinite queues using matrix-analytic methods. In: Proceedings of SMCtools 2008, Athens, Greece. ACM Press, New York (2008)
Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Performance Evaluation 64(9-12), 1153–1168 (2007)
van de Liefvoort, A.: The moment problem for continuous distributions. Technical report, University of Missouri, WP-CM-1990-02, Kansas City (1990)
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Horváth, A., Rácz, S., Telek, M. (2009). Moments Characterization of Order 3 Matrix Exponential Distributions. In: Al-Begain, K., Fiems, D., Horváth, G. (eds) Analytical and Stochastic Modeling Techniques and Applications. ASMTA 2009. Lecture Notes in Computer Science, vol 5513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02205-0_13
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DOI: https://doi.org/10.1007/978-3-642-02205-0_13
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