** Next:** Computational complexity of DR,
** Up:** Approximations of dimensionality reduction
** Previous:** Dimensionality reduction with partial
** Contents**

##

Dimensionality reduction with complete independence assumption

DR-CI (DR with complete independence assumption) ignores not only the
dependency that the length of the sojourn time in levels
has on how the Markov chain enters levels but also the dependency on
how it *exits* from levels (to level ).
Specifically, we assume that is independent of and .
Let for all and , and let
denote the process that is the same as except that
is replaced by for all and .
Figure 3.27(b) shows the Markov chain for the high priority
jobs (background process) that is obtained via DR-CI in the analysis
of an M/PH/2 queue with two priority classes. In DR-CI, the four
types of busy periods are represented by a *single* PH
distribution, ignoring the dependency that the duration of the busy
period has on the phase of the job in service at the beginning and end
of the busy period.

We choose such that
has the above two key properties
that has. The difference between and
lies in the dependencies in the sequence of the
sojourn times in levels . Observe that the sequence of the
sojourn times in levels is i.i.d. in
,
while it has some dependencies in .

More formally, the generator matrix
of
,
, is determined as follows.
Let be the -th moment of for .
We determine so that
and have
the same marginal -th moment of the sojourn time in levels :

We approximate
by a PH distribution,
,
matching the first three moments of ,
.
Let
as before.
Generator matrix
is then defined by

where
,
and is a row vector whose -th element is

** Next:** Computational complexity of DR,
** Up:** Approximations of dimensionality reduction
** Previous:** Dimensionality reduction with partial
** Contents**
Takayuki Osogami
2005-07-19