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Majorization and Extremal PH Distributions

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Matrix-Analytic Methods in Stochastic Models

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 27))

Abstract

This chapter presents majorization results for PH generators. Based on the majorization results, bounds on the moments and Laplace–Stieltjes transforms of phase-type distributions are found. Exponential distributions and Coxian distributions are identified to be extremal PH distributions with respect to all the moments and Laplace–Stieltjes transforms for certain subsets of PH distributions.

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References

  1. Aldous, D., Shepp, L.: The least variable phase type distribution is Erlang. Stoch. Models 3, 467–473 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23, 419–441 (1996)

    MATH  Google Scholar 

  3. Bodrog, L., Heindl, A., Horvath, A., Horvath, G., Telek, M.: Current results and open questions on PH and MAP characterization. In: Dagstuhl Seminar Proceedings 07461 (D. Bini, B. Meini, V. Ramaswami, M-A Remiche, and P. Taylor, eds.). Numerical Methods for Structured Markov Chains, http://drops.dagstuhl.de/opus/volltexte/2008/1401. (2008)

  4. Commault, C., Mocanu, S.: Phase-type distributions and representations: some results and open problems for system theory. Int. J. Control 76, 566–580 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. He, Q.M., Zhang, H.Q.: A note on unicyclic representations of phase type distributions. Stoch. Models 21, 465–483 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, Q.M., Zhang, H.Q.: Spectral polynomial algorithms for computing bi-diagonal representations for phase type distributions and matrix exponential distributions. Stoch. Models 22, 289–317 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. He, Q.M., Zhang, H.Q., Vera, J.C.: On some properties of bivariate exponential distributions. Stoch. Models 28, 187–206 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Horvath, A., Telek, M.: PhFit: A general purpose phase type fitting tool. In: Tools 2002, pp. 82–91. LNCS, vol. 2324. Springer, London (2002)

    Google Scholar 

  9. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  10. Maier, R.S.: The algebraic construction of phase-type distributions. Stoch. Models 7, 573–602 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic, New York (1979)

    MATH  Google Scholar 

  12. Minc, H.: Non-Negative Matrices. Wiley, New York (1988)

    Google Scholar 

  13. Neuts, M.F.: Probability distrubutions of phase type. In: Liber Amicorum Professor Emeritus H. Florin, pp. 173–206. Department of Mathematics, University of Louvain, Belgium (1975)

    Google Scholar 

  14. Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981)

    MATH  Google Scholar 

  15. O’Cinneide, C.A.: Characterization of phase-type distributions. Stoch. Models 6, 1–57 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. O’Cinneide, C.A.: Phase-type distributions and majorization. Ann. Appl. Probab. 1, 219–227 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. O’Cinneide, C.A.: Phase-type distributions: open problems and a few properties. Stoch. Models 15, 731–757 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sangüesa, C.: On the minimal value in Maier’s property concerning phase-type distributions. Stoch. Models 26, 124–140 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yao, R.H.: A proof of the steepest increase conjecture of a phase-type density. Stoch. Models 18, 1–6 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank reviewers for their valuable comments and suggestions. The authors would also like to thank Mr. Zurun Xu for proofreading the paper.

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Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Qi-Ming He

  2. University of Singapore, Singapore, 119245, Singapore

    Hanqin Zhang

  3. Tilburg University, Tilburg, The Netherlands

    Juan C. Vera

Authors
  1. Qi-Ming He
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  2. Hanqin Zhang
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  3. Juan C. Vera
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Corresponding author

Correspondence to Qi-Ming He .

Editor information

Editors and Affiliations

  1. , Faculté des Sciences, Univ. Libre de Bruxelles, Blvd. du Triomphe, CP 212, Bruxelles, 1050, Belgium

    Guy Latouche

  2. At&T Labs Research, Park Avenue 180, Florham Park, 07932, New Jersey, USA

    Vaidyanathan Ramaswami

  3. , IEOR, Columbia University, West 120th Street 500, New York City, 10027, New York, USA

    Jay Sethuraman

  4. , Dept of Industrial Engineering and OR, Columbia University, 312 Mudd Bldg., New York, 10027, New York, USA

    Karl Sigman

  5. IBM Thomas J. Watson Research Center, Kitchawan Road 1101, Yorktown Heights, 10598, New York, USA

    Mark S. Squillante

  6. , IEOR, Columbia University, West 120th Street 500, New York City, 10027, New York, USA

    David D. Yao

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He, QM., Zhang, H., Vera, J.C. (2013). Majorization and Extremal PH Distributions. In: Latouche, G., Ramaswami, V., Sethuraman, J., Sigman, K., Squillante, M., D. Yao, D. (eds) Matrix-Analytic Methods in Stochastic Models. Springer Proceedings in Mathematics & Statistics, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4909-6_6

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