Combining Adaptive and Standard Uniformization

1 Abstract Adaptive uniformization (AU) has recently been proposed as a method to compute transient measures in continuous-time Markov chains and has been shown to be especially attractive for solving large and sti dependability models. The major advantage of AU is that it requires at most as many iterations as standard uniformization (SU), and often far fewer, thus resulting in substantial computational savings. However, this computational gain can be o set by the need to compute more complex \jump probabilities" in AU, whose computation is more expensive than computing Poisson probabilities in SU. In particular, it can be shown that AU is computationally superior to SU if and only if the considered time instant is less than some threshold time value. To overcome this computational drawback, we combine AU and SU such that AU is used over the start of the time interval of interest, while SU is applied to the rest of the time interval. We show that combined AU/SU can be implemented in such a way that the combination introduces only minor computational overhead, the number of iterations required is almost as low as AU, and the cost of computing the jump probabilities is as low as SU. To demonstrate the bene ts of combined AU/SU, we apply it to a machine-repairman model, using a version of combined AU/SU implemented in the performance and dependability evaluation software package UltraSAN .