Example

Consider the QBD process shown in Figure 3.10(b), where

$$\mathbf{F}^{(\ell)} = \left(\begin{array}{cc} \lambda_1 & 0 ... ...\left(\begin{array}{cc} \mu & 0 \\ 0 & \mu \end{array}\right)$$

for $\ell\geq 1$, where $\sigma_1 = \lambda_1+\mu+\alpha_{12}$ and $\sigma_2 = \lambda_2+\mu+\alpha_{21}$. For this QBD process,

$$\begin{eqnarray*} \mbox{\boldmath {$\cal{F}$}}^{(\ell)}(x) & = & \left(\begin{a... ...}{\sigma_1} & 0\\ 0 & \frac{\mu}{\sigma_2} \end{array}\right) \end{eqnarray*}$$

for $\ell\geq 1$.