3.7.2 Explicit computation of ${\mathbf{G}}$

A special case of M/G/1-type processes occurs when ${\mathbf{B}}$ is a product of two vectors, i.e., ${\mathbf{B}}= \mbox{\boldmath {$\alpha$}}\cdot \mbox{\boldmath {$\beta$}}$. Assuming, without loss of generality, that $\mbox{\boldmath {$\beta$}}$ is normalized, then ${\mathbf{G}}= {\mathbf{1}}^T \cdot \mbox{\boldmath {$\beta$}}$, i.e., it is derived explicitly [77,78].

For this special case, ${\mathbf{G}}={\mathbf{G}}^n$, for $n \ge 1$. This special structure of matrix ${\mathbf{G}}$ simplifies the form of matrices $\widehat{{\mathbf{S}}}^{(i)}$ for $i \geq 1$, and $\S^{(i)}$ for $i \geq 0$ defined in Eq.(3.27):

$$\begin{array}{l c} \widehat{{\mathbf{S}}}^{(i)} = \widehat{{... ...bf{G}},~~i\geq 0, & {\mathbf{F}}^{(0)} \equiv \L . \end{array}$$ (3.29)

In this special case, ${\mathbf{G}}$ does not need to be either computed or fully stored, which is a considerable gain since in an M/G/1-type process computation of $G$ is expensive and $G$ needs to be stored throughout the solution procedure.