Definition of GFB process

The GFB process is a generalization of the FB process, and is a 2D Markov chain having less restrictions on its structure than the FB process. Specifically, we allow the background and foreground process to have complex interactions while the background process is in levels $<\kappa$ (and as a result, the GFB process behaves like a QBD process when the background process is in levels $<\kappa$), but the GFB process behaves like an FB process when the background process is in levels $\geq\kappa$ (see Figure 3.17). Recall that, in the FB process, the background process is a fixed QBD process, and only the behavior of the foreground process is affected by the level of the background process while the background process is in levels $<\kappa$.

Figure 3.17: The structure of the GFB process.

Essential restrictions in the GFB process are that

• There exists a level, $\kappa$, in the background process such that each of the background and foreground processes is a fixed QBD process while the background process is in levels $\geq\kappa$.
• There is no transition from level $<\kappa$ to level $>\kappa$ (i.e., any transition from level $<\kappa$ to level $\geq\kappa$ is to level $\kappa$) in the background process.

Given a 2D Markov chain, if one can identify the background and foreground process that constitute the 2D Markov chain and that satisfy the above restrictions, the 2D Markov chain can be reduced to a 1D Markov chain via DR. However, the resulting 1D Markov chain may not be analyzed efficiently. Thus, additionally, we require that

• Any transition in the 2D Markov chain changes the level of the foreground process by at most one.

If a 2D Markov chain satisfy the above two requirements, the 2D Markov chain can be reduced to a 1D Markov chain via DR, and the 1D Markov chain is a QBD process. We call such a 2D Markov chain a GFB process.