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##

Constructing a 1D Markov chain using an approximate background process

In Section 3.5.2, we obtain the generator matrix of an approximate
background process ,
.
In this section, we derive the generator matrix of the 1D Markov chain
reduced from the FB process by replacing the background process by .
We first introduce notations for the generator matrices of the foreground process
conditioned on the level of the background process.
Let
be the generator matrix of the foreground process, ,
given that the background process is in level .
As the foreground process is a QBD process given the level of ,
is of the form

for
. Recall that the generator matrix of the foreground process
when the background process is in levels is the same as
.
The 1D Markov chain reduced from the FB process is also a QBD process, and
its generator matrix, , is given by

where

for each ,
where is an identity matrix of order equal to the number of states in level
of for
,
is an identity matrix of order equal to the number of states
corresponding to the collection of PH distributions in
(i.e., when the busy period, the sojourn time in levels , is approximated by
PH distributions each with phases,
is an identity matrix),
and
is the number of states in level
of for .

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Takayuki Osogami
2005-07-19