Definition of FB process

Intuitively, an FB process consists of a background (QBD) process and a foreground (QBD) process, where the behavior (infinitesimal generator) of the foreground process can depend on the level of the background process. Additionally, we require that there exists a level, $\kappa$, in the background process such that the infinitesimal generator of the foreground process does not change while the background process is in levels $\geq\kappa$.

Consider a simpler case where the foreground and background processes are homogeneous birth-and-death processes. The background process, $B$, has a fixed generator matrix, $\mathbf{Q}_B$ (see Figure 3.13(a)):

$$\mathbf{Q}_B = \left(\begin{array}{cccc} -f_B & f_B & & \\ ... ...(b_B+f_B) & \ddots \\ & & \ddots & \ddots \end{array}\right).$$

On the other hand, the generator matrix of the foreground process, $F$, depends on the level of the background process. That is, when process $B$ is in level $d$ (i.e., when the state of $B$ is $d$), the process $F$ evolves according to the generator matrix $\mathbf{Q}_{F,d}$ (see Figure 3.13(b)):

$$\mathbf{Q}_{F,d} = \left(\begin{array}{cccc} -f_{F,d} & f_{F... ...+f_{F,d}) & \ddots \\ & & \ddots & \ddots \end{array}\right).$$

The FB process assumes that there exists a level, $\kappa$, of the background process such that $\mathbf{Q}_{F,d} = \mathbf{Q}_{F,\kappa}$ for all $d>\kappa$. Figure 3.13(c) shows the FB process consisting of the foreground and background processes in Figures 3.13(a)-(b).

Figure 3.13: FB process consisting of a foreground birth-and-death process and a background birth-and-death process.

(a) Background process

(b) Foreground process when the background process is in level $d$

(c) FB process

Recall the 2D Markov chains shown in Figure 3.1. These Markov chains are FB processes. In Figure 3.1(a), the background process tracks the number of donor jobs, and the foreground process tracks the number of beneficiary jobs. In Figure 3.1(b), the background process tracks the number of high priority jobs, and the foreground process tracks the number of low priority jobs. Also, recall the 2D Markov chain shown in Figure 3.4(c). This 2D Markov chain is an FB process, where the background process is a QBD process (and not a birth-and-death process). Again, the background process tracks the number of high priority jobs, and the foreground process tracks the number of low priority jobs.

In general, an FB process is defined by a vector of generator matrices $(\mathbf{Q}_B,(\mathbf{Q}_{F,0},...,\mathbf{Q}_{F,\kappa}))$. Here, $\mathbf{Q}_{F,d}$ denotes the generator matrix of the foreground process when the background process is in level $d$. We require that $\mathbf{Q}_B$ and $\mathbf{Q}_{F,d}$ satisfy the following characteristics:

• $\mathbf{Q}_B$ is a generator matrix for a QBD process; i.e., the background process is a QBD process.
• $\mathbf{Q}_{F,d} = \mathbf{Q}_{F,\kappa}$ for all $d>\kappa$; i.e., the infinitesimal generator of the foreground process stays the same while the background process is in levels $\geq\kappa$.
• $\mathbf{Q}_{F,d}$ is a generator matrix for a QBD process for $0\leq d \leq \kappa$; i.e., given the level of the background process, the foreground process is a QBD process.
• The order of the submatrix $\mathbf{L}_{F,d}^{(\ell)}$ is the same for all $d$, for each $\ell$; i.e., the state space of the foreground process is fixed.