License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.07461.14
URN: urn:nbn:de:0030-drops-13922
URL: https://drops.dagstuhl.de/opus/volltexte/2008/1392/
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Gr├╝nbaum, Alberto F.

QBD processes and matrix orthogonal polynomilas: somw new explicit examples

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07461.GrunbaumAlberto.Paper.1392.pdf (0.4 MB)


Abstract

In the case of birth-and-death processes
there are a few exactly solvable
situations where the n-step transition
matrix can be written down using the
Karlin-McGregor formula. A few of these
come from group representation theory.

I plan to show how this can be extended
to some instances of QBD processes with an
arbitrary finite number of phases. The group
involved is the set of all unitary
matrices of size N. For a fixed N one gets
examples where the number of phases is
a free parameter, and there are a few extra
parameters to play with. By tunning
these parameters
one can exhibit examples where states
are recurrent or transient.

The rather surprising fact that for these
examples one can compute everything
explicitly raises the issue of finding
a possible network application
for this piece of
mathematics that involves matrix
valued orthogonal polynomials.
I will give an ab-initio discussion of
the examples starting with the case of one phase.


BibTeX - Entry

@InProceedings{grunbaum:DagSemProc.07461.14,
  author =	{Gr\"{u}nbaum, Alberto F.},
  title =	{{QBD processes and matrix orthogonal polynomilas: somw new explicit examples}},
  booktitle =	{Numerical Methods for Structured Markov Chains},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{7461},
  editor =	{Dario Bini and Beatrice Meini and Vaidyanathan Ramaswami and Marie-Ange Remiche and Peter Taylor},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2008/1392},
  URN =		{urn:nbn:de:0030-drops-13922},
  doi =		{10.4230/DagSemProc.07461.14},
  annote =	{Keywords: QBD, orthogonal polynomials, Karlin-McGregor formula, representation theory}
}

Keywords: QBD, orthogonal polynomials, Karlin-McGregor formula, representation theory
Collection: 07461 - Numerical Methods for Structured Markov Chains
Issue Date: 2008
Date of publication: 07.04.2008


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