Constructing a 1D Markov chain using an approximate background process

In Section 3.5.2, we obtain the generator matrix of an approximate background process $\widetilde B$, $\mathbf{Q}_{\widetilde B}$. In this section, we derive the generator matrix of the 1D Markov chain reduced from the FB process by replacing the background process $B$ by $\widetilde B$.

We first introduce notations for the generator matrices of the foreground process conditioned on the level of the background process. Let $\mathbf{Q}_{d}$ be the generator matrix of the foreground process, $F$, given that the background process is in level $d$. As the foreground process is a QBD process given the level of $B$, $\mathbf{Q}_{d}$ is of the form

$$\mathbf{Q}_d = \left(\begin{array}{cccc} \mathbf{L}_d^{(0)}&... ...f{L}_d^{(2)}& \ddots\\ & &\ddots & \ddots \end{array}\right)$$

for $0\leq d \leq \kappa$. Recall that the generator matrix of the foreground process when the background process is in levels $>\kappa$ is the same as $\mathbf{Q}_\kappa$.

The 1D Markov chain reduced from the FB process is also a QBD process, and its generator matrix, $\mathbf{Q}$, is given by

$$\mathbf{Q} = \left(\begin{array}{cccc} \mathbf{L}^{(0)}& \ma... ...bf{L}^{(2)}& \ddots\\ & &\ddots & \ddots \end{array}\right),$$

where

$$\begin{eqnarray*} \mathbf{F}^{(\ell)} & = & \left(\begin{array}{cccc} \mathbf{I}... ...ay}\right) + \mathbf{Q}_{\widetilde B} \otimes \mathbf{I'}_\ell, \end{eqnarray*}$$

for each $\ell$, where $\mathbf{I}_i$ is an identity matrix of order equal to the number of states in level $\ell$ of $B$ for $1\leq i\leq \kappa-1$, $\mathbf{I}_\kappa$ is an identity matrix of order equal to the number of states corresponding to the collection of PH distributions in $\widetilde B$ (i.e., when the busy period, the sojourn time in levels $\geq\kappa$, is approximated by $n$ PH distributions each with $p$ phases, $\mathbf{I}_\kappa$ is an $np\times np$ identity matrix), and $\mathbf{I'}_\ell$ is the number of states in level $\ell$ of $F$ for $\ell\geq 0$.