Definition of RFB process

The RFB process generalizes the FB process to higher dimension. Intuitively, an RFB process consists of $m$ (QBD) processes, where the infinitesimal generator of the $i$-th process can depend on the level of the $(i-1)$-th process for $2\leq i\leq m$. Additionally, we require that there exists a level, $\kappa_{i-1}$, in the $(i-1)$-th process such that the generator matrix of the $i$-th process does not change while the $(i-1)$-th process is in levels $\geq \kappa_{i-1}$.

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

$$(\mathbf{Q}_{1}, (\mathbf{Q}_{2,0},...,\mathbf{Q}_{2,\kappa_... ..._2}), ...,(\mathbf{Q}_{m,0},...,\mathbf{Q}_{m,\kappa_{m-1}})).$$

Here, $\mathbf{Q}_{i,d}$ denotes the generator matrix of the $i$-th process when the $(i-1)$-th process is in level $d$. We require that $\mathbf{Q}_m$ satisfy the following characteristic:

• $\mathbf{Q}_1$ is a generator matrix for a QBD process; i.e., the first process is a QBD process.

Also, we require that $\mathbf{Q}_{i,d}$ satisfy the following characteristics for $2\leq i\leq m$:

• $\mathbf{Q}_{i,d} = \mathbf{Q}_{i,\kappa_{i-1}}$ for all $d>\kappa_{i-1}$; i.e., the infinitesimal generator of the $i$-th process stays the same while the $(i-1)$-th process is in levels $\geq \kappa_{i-1}$.
• $\mathbf{Q}_{i,d}$ is a generator matrix for a QBD process for $0\leq d \leq \kappa_{i-1}$; given the level of the $(i-1)$-th process, the $i$-th process is a QBD process.
• The order of the submatrix $\mathbf{L}_{i,d}^{(\ell)}$ is the same for all $d$, for each $\ell$; i.e., the state space of the $i$-th process is fixed.