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##

Analysis of the RFB process

The stationary probabilities in the RFB
process can be analyzed by applying the analysis in Section 3.5.2
recursively. Recall that, in the RFB process, the infinitesimal
generator of the -th process depend on the level of the -th
process for .
In Section 3.5.2, we analyzed the case of (the FB process).

We argue, by induction, that all the processes that constitute the RFB process
can be approximated by finite-phase QBD processes (1D Markov chains) via the approach in
Section 3.5.2. Then, the stationary probabilities in the
processes can be obtained by analyzing the stationary
probabilities in the 1D Markov chains. By our assumption,
the first process is a finite-phase QBD process that does not depend
on other processes, which proves the base case. Suppose that the
-th process is approximated by a QBD process with a finite number
of *phases*, . The QBD process typically has an
infinite number of *levels*. However, by the analysis in
Section 3.5.2, can be approximated by a QBD process
with a finite number of *levels*,
, such that
and
have stochastically similar effect on the
-th process, . Now, using
, process
can be approximated by a QBD process with a finite number of
*phases*. This completes our argument.

** Next:** Analysis of the GFB
** Up:** Dimensionality reduction
** Previous:** Analysis of the FB
** Contents**
Takayuki Osogami
2005-07-19