This paper calculates transient distributions of a special class of Markov processes with continuous state space and in continuous time, up to an explicit error bound. We approximate specific queues on \(\mathbb {R}\) with one-sided Lévy input, such as the M/G/1 workload process, with a finite-state Markov chain. The transient distribution of the original process is approximated by a distribution with a density which is piecewise constant on the state space. Easy-to-calculate error bounds for the difference between the approximated and actual transient distributions are provided in the Wasserstein distance. Our method is fast: to achieve a practically useful error bound, it usually requires only a few seconds or at most minutes of computation time.