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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 731261, 7 pages
http://dx.doi.org/10.1155/2015/731261
Research Article

On the Relation between Phase-Type Distributions and Positive Systems

Department of Computer Engineering, Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon 305-764, Republic of Korea

Received 21 January 2015; Revised 25 March 2015; Accepted 4 April 2015

Academic Editor: Shawn X. Wang

Copyright © 2015 Kyungsup Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. Benvenuti, L. Farina, and B. D. O. Anderson, “Filtering through combination of positive filters,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 46, no. 12, pp. 1431–1440, 1999. View at Publisher · View at Google Scholar · View at Scopus
  2. L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  3. T. Kaczorek and L. Sajewski, The Realization Problem for Positive and Fractional Systems, vol. 1, Springer, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. Kim, “A construction method for positive realizations with an order bound,” Systems & Control Letters, vol. 61, no. 7, pp. 759–765, 2012. View at Publisher · View at Google Scholar · View at Scopus
  5. K. Kim, “A constructive positive realization with sparse matrices for a continuous-time positive linear system,” Mathematical Problems in Engineering, vol. 2013, Article ID 878146, 9 pages, 2013. View at Publisher · View at Google Scholar
  6. A. Cumani, “On the canonical representation of homogeneous markov processes modelling failure-time distributions,” Microelectronics Reliability, vol. 22, no. 3, pp. 583–602, 1982. View at Publisher · View at Google Scholar · View at Scopus
  7. G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, vol. 43, SIAM, 1999.
  8. M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Dover Publications, New York, NY, USA, 1994. View at MathSciNet
  9. C. Commault and S. Mocanu, “Phase-type distributions and representations: some results and open problems for system theory,” International Journal of Control, vol. 76, no. 6, pp. 566–580, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. Carlson, “A note on M-matrix equations,” Journal of the Society for Industrial & Applied Mathematics, vol. 11, no. 4, pp. 1027–1033, 1963. View at Google Scholar
  12. D. Hershkowitz and H. Schneider, “Solutions of Z-matrix equations,” Linear Algebra and Its Applications, vol. 106, pp. 25–38, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. Schneider, “The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey,” Linear Algebra and Its Applications, vol. 84, pp. 161–189, 1986. View at Publisher · View at Google Scholar
  14. S. Muratori and S. Rinaldi, “Excitability, stability, and sign of equilibria in positive linear systems,” Systems & Control Letters, vol. 16, no. 1, pp. 59–63, 1991. View at Publisher · View at Google Scholar · View at Scopus
  15. P. De Leenheer and D. Aeyels, “Stabilization of positive linear systems,” Systems & Control Letters, vol. 44, no. 4, pp. 259–271, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. G. James and V. Rumchev, “Stability of positive linear discrete-time systems,” Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 53, no. 1, pp. 1–8, 2005. View at Google Scholar
  17. Y. Ohta, H. Maeda, and S. Kodama, “Reachability, observability, and realizability of continuous-time positive systems,” SIAM Journal on Control and Optimization, vol. 22, no. 2, pp. 171–180, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. M. E. Valcher, “Reachability properties of continuous-time positive systems,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1586–1590, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. C. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  20. J. Victory, “On nonnegative solutions of matrix equations,” Journal on Algebraic and Discrete Methods, vol. 6, no. 3, pp. 406–412, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  21. A. Bobbio, A. Horvath, M. Scarpa, and M. Telek, “Acyclic discrete phase type distributions: properties and a parameter estimation algorithm,” Performance Evaluation, vol. 54, no. 1, pp. 1–32, 2003. View at Publisher · View at Google Scholar · View at Scopus