6.7 Generation of the new ``upper'' process

The connectivity of with coupled with the existence of the gate state ensures that there is an efficient way to obtain the stochastic complement of by applying Lemma 2.8.2. The state space of the new process is . The infinitesimal generator of this new stochastic process is:

(6.10) |

Since is a single entry state for , all transitions from the states
in to are ``folded back'' into .
Let

where represents the portion of corresponding to transitions from to . is a matrix of zeros except for the column, which is obtained by applying Eq.(2.31) and Lemma 2.8.2:

where 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 is not required: since is a matrix that contains only one nonzero column that is added to the column of , 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 conditioned on the original process being in any of these states. These conditional probabilities are needed to formulate the pseudo-stochastic complement of according to Theorem 6.2, as explained in the following section.