New uniformization methods with steady-state detection

Properly implemented, uniformization methods are numerically stable. The article presents new uniformization methods with steady-state detection for computing with bounded approximation error and small impact of roundoff errors: (1) the transient probability vector and the averaged transient probability vector of a continuous-time Markov chain, (2) the expected transient reward rate and the expected averaged reward rate of a Markov reward model with reward rates associated to states. To develop the new methods, we use Semal's results on monotone iterative methods for computing with error bounds the steady-state probability vector of the subordinated discrete-time Markov chain. The methods are the first uniformization methods exploiting steady-state detection with strictly bounded approximation error for arbitrary finite continuous-time Markov chains and can be expected to reduce the computational cost of uniformization methods when no component of the steady-state probability vector of the continuous-time Markov chain is tiny and the transient probability vector of the continuous time Markov chain reaches steady-state long before the largest time at which the quantity of interest is to be computed. The numerical stability of the new methods is argued by measuring relative roundoff errors and is found to be excellent.