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# 6.8 Generation of the new lower'' process

After applying Theorem 6.2, the infinitesimal generator of the new process with state space is given by:

 (6.11)

Matrices for , , , and are defined in Section 6.5. contains the entries of that correspond to transitions from to . What remains to be defined are the matrices , , and .

Since all interaction of the lower'' process with states in is done through , it follows that only the rates out of need to be altered. Indeed, by applying Theorem 6.2 we see:

and

The new terms and are introduced by pseudo-stochastic complementation and represent how states from and enter the new process, respectively.

Let be conditional stationary distribution of . This can be derived by normalizing the stationary distribution of the stochastic complement of . Since the stochastic complement of the upper'' process is solved with the matrix geometric method, it follows that

Recall that the application of pseudo-stochastic complementation adds only a component to the first row-block of . Since we allow transitions from any level in to all levels in , matrix can be full:
 (6.12)

Matrix represents the communication pattern of states in to states in and it is a matrix of zeros except for a finite portion of its row that corresponds to entries from to states in . Let be a scalar that represents the rate with which state is left to enter and can be defined as the sum of all entries on the row of .

Next, we compute . From Eq.(6.12), it follows that:

 (6.13)

 (6.14)

 (6.15)

Let , therefore

 (6.16)

The term

is bounded by a constant, because of the nature of the infinitesimal generator defined in Eq.(6.1). The is strictly less than the , since it is a portion of the positive sum This results in a finite value of .

Furthermore, we define

This vector sums to one (because of the normalization) and can be partitioned according to the levels of as:

It follows that

and

Therefore, the sum 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 and to converge. converges by definition (see the definition of in Eq.(6.1)). Similarly, converges since its component sums are finite. By the definition of in Section 6, the sum is finite.

Next: 6.9 Multiple upper-lower classes Up: 6. Aggregate Solutions of Previous: 6.7 Generation of the
Alma Riska 2003-01-13