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#

State of the art in the analysis of multidimensional Markov chains

As multidimensional Markov chains are prevalent in the analysis of
multiserver systems, several approaches have been proposed for
their analysis. A most popular approach is
to simply truncate the state space of the Markov chain and analyze the
resulting Markov chain, for example, via matrix analytic methods. By
contrast, DR
reduces a multidimensional Markov chain into a 1D Markov chain, but
the state space is not simply truncated. Rather, the infinite
portion of the state space is aggregated into a smaller space,
capturing detailed behavior (such as the first three moments and
correlations of the sojourn time distributions in the infinite
portion) of the original multidimensional Markov chain. There are
also approaches that can be applied to multidimensional Markov chains
without state space truncation; however, these approaches are limited
in the class of Markov chains to which they can be applied either due
to essential restrictions or due to computational complexity.
We will see that these approaches have limitations in the analysis
of the RFB and GFB processes.

**Subsections**

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Takayuki Osogami
2005-07-19