Solution of transient markovian state probabilities and rate of approach to equilibrium states

The differential equations for transient state probabilities for Markovian processes are examined to derive the rate of convergence of transient states to equilibrium states. There is an acute need to solve the balance equations for large states, particularly for handling computer per- formance modeling with a network of queues that do not satisfy product form solutions or can- not be cast into the forms convenient for mean value analysis. The rate of convergence to equilibrium states is derived for irreducible aperiodic homogeneous Markov chains on the basis of a geometrical interpretation. A numerical integration method with dynamic step-size adjustments is applied and compared against the power method of Wallace and Rosenberg.