Next: Examples of RFB processes Up: FB, RFB, and GFB Previous: Definition of FB process   Contents

## Definition of RFB process

The RFB process generalizes the FB process to higher dimension. Intuitively, an RFB process consists of (QBD) processes, where the infinitesimal generator of the -th process can depend on the level of the -th process for . Additionally, we require that there exists a level, , in the -th process such that the generator matrix of the -th process does not change while the -th process is in levels .

More formally, an RFB process consisting of processes is characterized by parameters

Here, denotes the generator matrix of the -th process when the -th process is in level . We require that satisfy the following characteristic:
• is a generator matrix for a QBD process; i.e., the first process is a QBD process.
Also, we require that satisfy the following characteristics for :
• for all ; i.e., the infinitesimal generator of the -th process stays the same while the -th process is in levels .
• is a generator matrix for a QBD process for ; given the level of the -th process, the -th process is a QBD process.
• The order of the submatrix is the same for all , for each ; i.e., the state space of the -th process is fixed.

Next: Examples of RFB processes Up: FB, RFB, and GFB Previous: Definition of FB process   Contents
Takayuki Osogami 2005-07-19