6.7 Generation of the new ``upper'' process

The connectivity of ${\cal{U}}$ with ${\cal{L}}$ coupled with the existence of the gate state $g$ ensures that there is an efficient way to obtain the stochastic complement of ${\cal{U}}$ by applying Lemma 2.8.2. The state space of the new process is ${\cal{G}}^+_g \cup {\cal{U}}$. The infinitesimal generator of this new stochastic process is:

$${\mathbf{Q}}_{{\cal{G}}^+_g \cup {\cal{U}}} = \left[ \begin... ...dots & \vdots & \vdots & \vdots & \ddots \end{array}\right] .$$ (6.10)

Matrices ${\mathbf{F}}^{(1)}_{{\cal{U}}{\cal{U}}}$, $\L _{{\cal{U}}{\cal{U}}}$, and ${\mathbf{B}}^{(j)}_{\cal{UU}}$, $j \ge 1$, are defined in Section 6.5. $\L ^{(0)}_{{\cal{G}}^+_g{\cal{G}}^+_g}$ contains the rows and columns of $\L ^{(0)}$ corresponding to states in ${\cal{G}}^+_g$, except that the diagonal element $\L ^{(0)}_{{\cal{G}}^+_g{\cal{G}}^+_g}[g,g]$ might differ from the corresponding one in $\L ^{(0)}$, since any transition from $g$ to ${\cal{L}}$ is rerouted back to $g$ itself, hence ignored. $\widehat{{\mathbf{F}}}^{(1)}_{{\cal{G}}^+_g{\cal{U}}}$ contains the entries of $\widehat{{\mathbf{F}}}^{(1)}$ that represent the transitions from states in ${\cal{G}}^+_g$ to ${\cal{U}}$. What remains to be defined are the matrices $\widehat{{\mathbf{B}}}^{(j0)}_{{\cal{U}}{\cal{G}}^+_g}$, $j \ge 1$.

Since $g$ is a single entry state for ${\cal{U}}$, all transitions from the states in ${\cal{U}}$ to ${\cal{L}}$ are ``folded back'' into $g$. Let

$$\widehat{{\mathbf{B}}}^{(j0)}_{{\cal{U}}{\cal{G}}^+_g} = \wi... ...{\cal{G}}^+_g} + \overline{{\mathbf{B}}}^{(j)}, ~~~ j \geq 1 ,$$

where $\widehat{{\mathbf{B}}}^{(j)}_{{\cal{U}}{\cal{G}}^+_g}$ represents the portion of $\widehat{{\mathbf{B}}}^{(j)}_{{\cal{U}}}$ corresponding to transitions from ${\cal{U}}$ to ${\cal{G}}^+_g$. $\overline{{\mathbf{B}}}^{(j)}$ is a matrix of zeros except for the $g^{\rm th}$ column, which is obtained by applying Eq.(2.31) and Lemma 2.8.2:

$$\overline{{\mathbf{B}}}^{(j)}[{\cal{N}}_u,g] = \left( \sum_{... ...{\cal{U}}{\cal{L}}} \right) \cdot{\mathbf{e}} ~~~ j \geq 1,$$

where ${\mathbf{e}}$ is a row vector of one with the appropriate dimension.

The result is a GI/M/1-type process that is solved using the matrix-geometric method outlined in Section 3.1. Note that the computation of the sum $\sum_{j=1}^{\infty}\overline{{\mathbf{B}}}^{(j)}$ is not required: since $\overline{{\mathbf{B}}}^{(j)}$ is a matrix that contains only one nonzero column that is added to the $g^{\rm th}$ column of $\widehat{{\mathbf{B}}}^{(j)}_{{\cal{U}}{\cal{G}}^+_g}$, we opt to discard this particular column when using Eq.(3.3).

After applying the matrix-geometric method, we obtain an expression for the stationary probabilities of the states in ${\cal{G}}^+_g \cup {\cal{U}}$ conditioned on the original process being in any of these states. These conditional probabilities are needed to formulate the pseudo-stochastic complement of ${\cal{G}}^-_g \cup {\cal{L}}$ according to Theorem 6.2, as explained in the following section.