Direct Methods - GTH and Block GTH

The GTH algorithm is a steady-state direct method that has nice properties. Basically, the algorithm works by eliminating states, one at a time from the set of states of the model. In other words, from the initial transition probability matrix, the stochastic complement of the matrix for a state is obtained and, at each subsequent step, a new stochastic complement is calculated from that found in the preceding step. Finally, an upper-triangular matrix $U$ is obtained and the system $\pi = \pi U$ is solved.

The block version of the algorithm is also implemented. In this case, blocks of states are eliminated at a time, instead of a single state. As a consequence, the procedure requires calculation of the inverse of the diagonal blocks of the uniformized stochastic matrix.

One advantage of the method is that is does not require any subtractions. Another interesting advantage is that the approach has a probabilistic interpretation. One problem is the ``fill-in'' that may occur in the matrix, which can destroy the sparseness of the original. However the method does preserve an existing banded structure (which is common in many system models).

To use this method, choose the button Analytical Model Solution. Choose Stationary $\rightarrow$ Exact. The interface is shown in Figure .

Figure: The Stationary Exact Methods.

NOTE: If the method to be used is the block, we must specify the final state and the block size (the number of states in the block). In chapter  we show how the user can specify the blocks after visualizing the transition probability matrix.