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Dimensionality reduction with partial independence assumption
DRPI (DR with partial independence assumption) ignores the dependency
that the sojourn time in levels has on how
the Markov chain enters levels (from level ).
Specifically, we assume that is independent of
. Let
for all for each , and let
denote the process that is the same as
except that is replaced by for all for each .
Figure 3.27(a) shows the Markov chain for
the high priority jobs (background process) that we obtain via DRPI in the analysis of an
M/PH/2 queue with two priority classes.
(Recall the Markov chain that we obtain via DR, Figure 3.5 in
Section 3.1, where
the four types of busy periods are
approximated by four PH distributions, respectively.)
In DRPI, the four types of busy
periods are represented by two PH distributions, ignoring the
dependency that the duration of the busy period has on the phase of
the job in service at the beginning of the busy period.
Figure 3.27:
Background processes on a finite state space, obtained via (a) DRPI and (b) DRCI.
Labels on the transitions in the busy periods are omitted for clarity.

In general, we require that process has the following
two key properties:
 The probability of event in
is the same as that in (and that in as well).
 We choose such that the marginal distribution of
the sojourn time in levels in wellrepresents that in (and hence in as well).
Hence, and would have
stochastically the same total sojourn time in levels in the long run
if the second property did not involve the approximation (i.e., if the marginal distribution is fitted exactly).
However,
and have different
autocorrelation in the sequence of the sojourn times in levels .
More formally, the generator matrix
of ,
, is determined as follows.
Let
be the th moment of for
for each .
We determine
so that and have
the same marginal th moment of the sojourn time in levels :
where
denotes the stationary probability vector that is in level ,
which can be calculated via matrix analytic methods as in Section 3.2.
We approximate
by a PH distribution,
,
matching the first three moments of ,
.
Let
as before.
Generator matrix
is then defined by
where
,
, and
are submatrices of as defined in Section 3.5.2,
Observe that the number of PH distributions used to approximate the
sojourn time distributions in levels is reduced from ,
in , to , in . The next approximation, DRCI, uses only one PH
distribution.
Next: Dimensionality reduction with complete
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Takayuki Osogami
20050719