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The RG-Factorizations in Stochastic Models

The RG-Factorizations in Stochastic Models. Dr. Quan-Lin Li Department of Industrial Engineering Tsinghua University Beijing 100084, P.R. China. Outline of this talk. Why to need the RG-factorizations How to construct the RG-factorizations How to apply the RG-factorizations

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The RG-Factorizations in Stochastic Models

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  1. The RG-Factorizations in Stochastic Models Dr. Quan-Lin Li Department of Industrial Engineering Tsinghua University Beijing 100084, P.R. China

  2. Outline of this talk • Why to need the RG-factorizations • How to constructthe RG-factorizations • How to apply the RG-factorizations • Promising issues in the future

  3. Our main problem is described as follows: Our Question: How to compute? Why to need • From 1996 to 2000, my research focuses on quasi-stationary distributions of stochastic models

  4. Why to need

  5. Why to need

  6. Why to need • Our work from 1996 to 1999 was to develop theLU-block-decomposition for Markov chains of M/G/1 type and GI/M/1 type • Our method is different from that used by Bean, Latouche, Taylor etc.

  7. Why to need

  8. The LU-block-decomposition is Our Question: Such a solution is OK? Why to need

  9. Why to need • For a special Markov chain, we obtained two differentLU-block-decompositions, which lead to two different expressions • One of them is correct and is the same as that in the literature; while another is wrong Why?

  10. Why to need • We analyzed many real examples and then found the main reasons • These computations motivate us to extend theLU-block-decomposition to the RG-factorization

  11. Why to need

  12. Why to need ? ?

  13. Our Comparisons • Utility of the RG-factorization is related to the classification of state by means of the diagonal matrix, and keep effective computations • Better than LU-decomposition

  14. How to construct

  15. How to construct

  16. The UL-type RG-factorization

  17. The UL-type RG-factorization

  18. The UL-type RG-factorization

  19. Some special cases

  20. Some special cases

  21. How to apply

  22. Remarks

  23. Infinite states Finite states Huge The UL-type RG-factorization A crucial advance Finite states Smaller

  24. The LU-type RG-factorization

  25. The LU-type RG-factorization

  26. The LU-type RG-factorization

  27. Some special cases

  28. How to apply

  29. UL-type RG-factorization LU-type RG-factorization Comparison for UL- and LU-type Question:Systems of linear equations

  30. Quasi-stationary distributions Tailed analysis Sensitive analysis Theory Markov Reward processes Markov Decision processes The RG-factorizations Evolutionary games Networking safety Applications Computer networks Production systems Real-time management Our work on the RG-factorizations

  31. Promising problems (1) • For the RG-factorizations: 1. It is interesting to consider the d-period for the R-, U- and G-measures. For example (1)A = A0 + A1 + A2 is irreducible and is d-period, the two matrices R and G are d-period? (2) For a Markov chain of GI/G/1 type, what happen to ? Such a work is useful for tailed analysis

  32. Promising problems (2) • For the RG-factorizations: 1.It is interesting to consider spectral analysis for the R-, U- and G-measures. When A = A0 + A1 + A2 is irreducible and is infinite size, how to analyze the spectral of the two matrices R and G ? 2. For a Markov chain of GI/G/1 type, what happen to ?

  33. Promising problems (3)

  34. Promising problems (4)

  35. Train repairable networks

  36. Local Optimization Local Optimization Information Theory Queueing Networks Real-time management Global Optimization

  37. Thanks for youand questions ?

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