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Dimensionality reduction with partial independence assumption
DR-PI (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 DR-PI 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 DR-PI, 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) DR-PI and (b) DR-CI.
Labels on the transitions in the busy periods are omitted for clarity.
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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
well-represents 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, DR-CI, uses only one PH
distribution.
Next: Dimensionality reduction with complete
Up: Approximations of dimensionality reduction
Previous: Approximations of dimensionality reduction
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Takayuki Osogami
2005-07-19