B. (B)MAP/PH/1 queues

A MAP/PH/1 represents a single server queue that has MAP arrival process and PH service process. The MAP/PH/1 queue is a quasi birth-death process whose matrices can be computed using the matrix parameters that define the MAP and the PH processes of the MAP/PH/1 queue. Let the MAP describtors to be $({\mathbf{D}}_0,{\mathbf{D}}_1)$ of order $m_A$ and the PH service time parameters to be $(\mbox{\boldmath {$\tau$}}, {\mathbf{T}})$ of order $m_B$. The infinitesimal generator matrix ${\mathbf{Q}}_{MAP/PH/1}$ for the corresponding MAP/PH/1 queue has a structure given by

$${\mathbf{Q}}_{MAP/PH/1} = \left[ \begin{array}{c c c c c c c... ...vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right] ,$$ (B.1)

where each of its matrices is defined as follows

$$\begin{array}{l l} \widehat{{\mathbf{L}}}= {\mathbf{D}}_0, ~... ...bf{F}}= {\mathbf{D}}_1 \otimes {\mathbf{I}}_{m_B}, \end{array}$$ (B.2)

where ${\mathbf{I}}_k$ is the order $k$ identity matrix, ${\mathbf{T}}^0 = - {\mathbf{T}}\cdot {\bf e}$, and $\otimes$ denotes the Kroneker product.

A BMAP/PH/1 represent a single server queue that has BMAP arrival process and PH service process. The BMAP/PH/1 queue results in a M/G/1-type process. Let the BMAP parameteres be ${\mathbf{D}}_i$ for $i \geq 0$ and PH parameters be $(\mbox{\boldmath {$\tau$}}, {\mathbf{T}})$ of order $m_A$ and $m_B$, respectively. The infinitesimal generator matrix ${\mathbf{Q}}_{BMAP/PH/1}$ for the corresponding BMAP/PH/1 queue is given by

$${\mathbf{Q}}_{BMAP/PH/1} = \left[ \begin{array}{c c c c c c ... ...ots & \vdots & \vdots & \vdots & \ddots \end{array} \right] ,$$ (B.3)

where each of its matrices is defined as follows

$$\begin{array}{l l} \widehat{{\mathbf{L}}}= {\mathbf{D}}_0, &... ...thbf{D}}_i \otimes {\mathbf{I}}_{m_B}, ~~~i \geq 1, \end{array}$$ (B.4)

where ${\mathbf{I}}_k$ is the order $k$ identity matrix, ${\mathbf{T}}^0 = - {\mathbf{T}}\cdot {\bf e}$, and $\otimes$ denotes the Kroneker product.