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#

FB, RFB, and GFB processes

In this section, we define the foreground-background (FB) process,
the recursive FB (RFB) process, and the generalized FB (GFB) process,
and provide examples of these processes.
In the rest of this chapter, we denote a matrix by a bold face letter such as
and its element by
, and we
use to denote a vector and to denote its
-th element. Also, we use matrix to denote the
generator matrix of a QBD process ``characterized by
parameter .'' Here, may be a single letter or number, denoting
process or the -th QBD process, or may be a pair of numbers
, denoting the -th QBD process of type . Unless otherwise stated, we express
using submatrices,
,
, and
, such that

where
encodes (local) transitions within level ,
encodes (forward) transitions from level to level
, and
encodes (backward) transitions from level to
level , for each (see Section 3.2).

**Subsections**

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** Up:** Dimensionality reduction of Markov
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Takayuki Osogami
2005-07-19