Steady-state analytical solvers

In a large number of applications, it is necessary to find the steady-state solution of the system being analyzed. Although the analyst is often interested in the system behavior during a finite time interval $(0,t)$, the steady-state solution may be a good approximation for values of $t$ ``sufficiently large''. From this solution, several measures of interest can be calculated, for instance the system availability, which is the fraction of time the system remains operational during $(0,\infty)$. Other examples of measures include the average number of tasks processed per unit of time (throughput), the expected time to process a task, etc.

We can divide the methods used to obtain the steady state solution into direct and iterative methods. A method is called direct when the exact solution is obtained after a finite number of steps. On the other hand, a method is called iterative when it produces a sequence of approximate solutions that converge to the exact value.

In general, direct methods are appropriate when the state space of the model is not very large and when the corresponding state transition matrix is not sparse. Iterative methods, on the other hand, are appropriate when the state transition matrix is large and sparse, since they preserve the sparseness of the matrix.

For Markovian models the direct methods implemented are GTH and block GTH, and the iterative methods are SOR, Jacobi, Gauss-Seidel, and Power.