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 $\mathbf{X}$ and its $(i,j)$ element by $(\mathbf{X})_{i,j}$, and we use $\vec{x}$ to denote a vector and $(\vec{x})_i$ to denote its $i$-th element. Also, we use matrix $\mathbf{Q}_Y$ to denote the generator matrix of a QBD process ``characterized by parameter $Y$.'' Here, $Y$ may be a single letter or number, denoting process $Y$ or the $Y$-th QBD process, or $Y$ may be a pair of numbers $(i,j)$, denoting the $i$-th QBD process of type $j$. Unless otherwise stated, we express $\mathbf{Q}_Y$ using submatrices, $\mathbf{L}_Y^{(h)}$, $\mathbf{F}_Y^{(h)}$, and $\mathbf{B}_Y^{(h)}$, such that

$$\mathbf{Q}_Y = \left(\begin{array}{lllll} \mathbf{L}_Y^{(0)... ...Y^{(2)} & \\ & & \ddots & \ddots & \ddots \end{array}\right),$$

where $\mathbf{L}_Y^{(h)}$ encodes (local) transitions within level $h$, $\mathbf{F}_Y^{(h)}$ encodes (forward) transitions from level $h$ to level $h+1$, and $\mathbf{B}_Y^{(h)}$ encodes (backward) transitions from level $h$ to level $h-1$, for each $h$ (see Section 3.2).