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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 .

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