6.8 Generation of the new ``lower'' process

After applying Theorem 6.2, the infinitesimal generator of the new process with state space ${\cal{G}}^-_g \cup {\cal{L}}$ is given by:

$${\mathbf{Q}}_{{\cal{G}}^-_g \cup {\cal{L}}} = \left[ \begin... ...dots & \vdots & \vdots & \vdots & \ddots \end{array}\right] ,$$ (6.11)

Matrices ${\mathbf{F}}^{(j)}_{{\cal{L}}{\cal{L}}}$ for $j \geq 1$, $\L ^{(1)}_{{\cal{L}}{\cal{L}}}$, $\L _{{\cal{L}}{\cal{L}}}$, and ${\mathbf{B}}^{(1)}_{{\cal{L}},{\cal{L}}}$ are defined in Section 6.5. $\widehat{{\mathbf{B}}}^{(1)}_{{\cal{L}}{\cal{G}}^-_g}$ contains the entries of $\widehat{{\mathbf{B}}}^{(1)}$ that correspond to transitions from ${\cal{G}}^-_g$ to ${\cal{L}}$. What remains to be defined are the matrices $\widehat{{\mathbf{F}}}^{(0j)}_{{\cal{G}}^-_g{\cal{L}}}$, $j \ge 1$, and $\L ^{(00)}_{{\cal{G}}^-_g{\cal{G}}^-_g}$.

Since all interaction of the ``lower'' process with states in ${\cal{U}}$ is done through ${\cal{G}}^-_g$, it follows that only the rates out of $g$ need to be altered. Indeed, by applying Theorem 6.2 we see:

$$\L ^{(00)}_{{\cal{G}}^-_g{\cal{G}}^-_g} = \L ^{(0)}_{{\cal{G}}^-_g{\cal{G}}^-_g} + \overline{{\mathbf{L}}}$$

and

$$\widehat{{\mathbf{F}}}^{(0j)}_{{\cal{G}}^-_g{\cal{L}}} = \wi... ...al{L}}{\cal{L}}} +\overline{{\mathbf{F}}}^{(j)}, ~~~ j \geq 1.$$

The new terms $\overline{{\mathbf{L}}}$ and $\overline{{\mathbf{F}}}^{(j)}$ are introduced by pseudo-stochastic complementation and represent how states from ${\cal{G}}^+$ and ${\cal{U}}$ enter the new process, respectively.

Let $\overline{\mbox{\boldmath {$\alpha$}}} = [\overline{\mbox{\boldmath {$\alpha$}}}^{(0)}, \overline{\mbox{\boldmath {$\alpha$}}}^{(1)}, \ldots]$ be conditional stationary distribution of ${\cal{G}}^+ \cup {\cal{U}}$. This can be derived by normalizing the stationary distribution of the stochastic complement of ${\cal{G}}^+_g \cup {\cal{U}}$. Since the stochastic complement of the ``upper'' process is solved with the matrix geometric method, it follows that

$$\overline{\mbox{\boldmath {$\alpha$}}}^{(j)} = \overline{\mb... ...ath {$\alpha$}}}^{(1)} \cdot {\mathbf{R}}^{(j-1)} ~~j \geq 1 .$$

Recall that the application of pseudo-stochastic complementation adds only a component ${\mathbf{Q}}{[{\cal{G}}^-_g\cup {\cal{L}},{\cal{G}}^+\cup{\cal{U}}]} \cdot{\mat... ...a$}}}\cdot{\mathbf{Q}}{[{\cal{G}}^+\cup{\cal{U}},{\cal{G}}^-_g\cup {\cal{L}}}])$ to the first row-block of ${\mathbf{Q}}_{{\cal{G}}^-_g\cup {\cal{L}}}$. Since we allow transitions from any level in ${\cal{U}}$ to all levels in ${\cal{L}}$, matrix ${\mathbf{Q}}[{\cal{G}}^+\cup{\cal{U}},{\cal{G}}^-_g\cup {\cal{L}}]$ can be full:

$${\mathbf{Q}}{[{\cal{G}}^+\cup{\cal{U}},{\cal{G}}^-_g\cup {\c... ...vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right] .$$ (6.12)

Matrix ${\mathbf{Q}}{[{\cal{G}}^-_g\cup {\cal{L}},{\cal{G}}^+\cup{\cal{U}}]}$ represents the communication pattern of states in ${\cal{G}}^-_g \cup {\cal{L}}$ to states in ${\cal{G}}^+ \cup {\cal{U}}$ and it is a matrix of zeros except for a finite portion of its $g^{th}$ row that corresponds to entries from $g$ to states in ${\cal{G}}^+ \cup {\cal{U}}^{(1)}$. Let $o$ be a scalar that represents the rate with which state $g$ is left to enter ${\cal{G}}^+ \cup {\cal{U}}$ and can be defined as the sum of all entries on the $g^{th}$ row of ${\mathbf{Q}}{[{\cal{G}}^-_g\cup {\cal{L}},{\cal{G}}^+\cup{\cal{U}}]}$.

Next, we compute ${\mathit{Norm}}(\overline{\mbox{\boldmath {$\alpha$}}} \cdot {\mathbf{Q}}{[{\cal{G}}^+\cup{\cal{U}},{\cal{G}}^-_g\cup {\cal{L}}]})$. From Eq.(6.12), it follows that:

$$\overline{\mbox{\boldmath {$\alpha$}}} \cdot {\mathbf{Q}}{[{... ... \cdot \widehat{{\mathbf{B}}}_{{\cal{U}}{\cal{G}}^-_g}^{(i)},$$ (6.13)

$$\overline{\mbox{\boldmath {$\alpha$}}} \cdot {\mathbf{Q}}{[{... ...a$}}}^{(i+1)} \cdot {\mathbf{B}}^{(i)}_{{\cal{U}}{\cal{L}}} ,$$ (6.14)

$$\overline{\mbox{\boldmath {$\alpha$}}} \cdot {\mathbf{Q}}{[{... ...i)} \cdot {\mathbf{B}}^{(i-j)}_{{\cal{U}}{\cal{L}}} ~~~ j > 1.$$ (6.15)

Let $T=\overline{\mbox{\boldmath {$\alpha$}}} \cdot {\mathbf{Q}}{[{\cal{G}}^+\cup{\cal{U}},{\cal{G}}^-_g\cup {\cal{L}}]} \cdot {\mathbf{e}}$, therefore

$$\begin{array}{l l} T= &\left( \overline{\mbox{\boldmath {$\... ...}}{\cal{L}}} \right) \right) \cdot {\mathbf{e}}. \end{array}$$ (6.16)

The term

$$\widehat{{\mathbf{B}}}^{(i)}_{{\cal{U}}{\cal{G}}^-_g} + \su... ...}} + \sum_{j=1}^\infty {\mathbf{F}}^{(j)}_{{\cal{U}}{\cal{L}}}$$

is bounded by a constant, because of the nature of the infinitesimal generator defined in Eq.(6.1). The ${\mathit{RowSum}}(\widehat{{\mathbf{B}}}^{(i)}_{{\cal{U}}{\cal{G}}^-_g} + \sum_{j=1}^{i-1} {\mathbf{B}}^{(j)}_{{\cal{U}}{\cal{L}}})$ is strictly less than the ${\mathit{RowSum}}(-\L -\sum_{j=1}^{\infty}{\mathbf{F}}^{(j)})$, since it is a portion of the positive sum $\widehat{{\mathbf{B}}}^{(i)}+{\mathbf{B}}^{(i-1)}+....+{\mathbf{B}}^{(1)}$ This results in a finite value of $T$.

Furthermore, we define

$$\mbox{\boldmath {$\rho$}} = {\mathit{Norm}}(\overline{\mbox{... ...}}{[{\cal{G}}^+\cup{\cal{U}},{\cal{G}}^-_g\cup {\cal{L}}]}}{T}$$

This vector sums to one (because of the normalization) and can be partitioned according to the levels of ${\cal{L}}$ as:

$$\mbox{\boldmath {$\rho$}}= [ ~\mbox{\boldmath {$\rho$}}^{(0)... ...ath {$\rho$}}^{(1)}, \mbox{\boldmath {$\rho$}}^{(2)}, \ldots].$$

It follows that

$$\overline{{\mathbf{L}}}= o\cdot \mbox{\boldmath {$\rho$}}^{(0)}$$

and

$$\overline{{\mathbf{F}}}^{(j)} = o\cdot \mbox{\boldmath {$\rho$}}^{(j)}~~~j\geq 1 .$$

Therefore, the sum $\sum_{j=1}^\infty\overline{{\mathbf{F}}}^{(j)} = o \cdot \sum_{j=1}^\infty \mbox{\boldmath {$\rho$}}^{(j)}$ converges.

The new process, defined by Eq.(6.11) is ergodic and can be solved using the M/G/1 algorithm outlined in Subsection 3.4. But, in order to apply the algorithm, it is necessary for the sums $\sum_{j=1}^{\infty} {\mathbf{F}}^{(j)}_{{\cal{L}}{\cal{L}}}$ and $\sum_{j=1}^{\infty} \widehat{{\mathbf{F}}}^{(0j)}_{{\cal{G}}^-_g{\cal{L}}}$ to converge. $\sum^{\infty}_{j=1} {\mathbf{F}}^{(j)}_{{\cal{L}}{\cal{L}}}$ converges by definition (see the definition of ${\mathbf{Q}}$ in Eq.(6.1)). Similarly, $\sum^{\infty}_{j=1} \widehat{{\mathbf{F}}}^{(0j)}_{{\cal{G}}^-_g{\cal{L}}}$ converges since its component sums are finite. By the definition of ${\mathbf{Q}}$ in Section 6, the sum $\sum^{\infty}_{j=1}\widehat{{\mathbf{F}}}^{(j)}_{{\cal{L}}{\cal{L}}}$ is finite.