The randomization technique for computing transient probabilities of Markov processes is presented. The technique is applied to a Markov process model of a simplified fault tolerant computer system for illustrative purposes. It is applicable to much larger and more complex models. Transient state probabilities are computed, from which reliabilities are derived. An accelerated version of the randomization algorithm is developed which exploits ''stiffness' of the models to gain increased efficiency. A great advantage of the randomization approach is that it easily allows probabilities and reliabilities to be computed to any predetermined accuracy.