A technique is developed for comparing a non-Markov process to a Markov process on a general state space with many possible stochastic orderings. Two such comparisons with a common Markov process yield a comparison between two non-Markov processes. The technique, which is based on stochastic monotonicity of the Markov process, yields stochastic comparisons of the limiting distributions and the marginal distributions at single time points, but not the joint distributions. These stochastic comparisons are obtained under the condition that the one-step transition probabilities (in discrete time) or infinitesimal transition rates (in continuous time) are appropriately ordered, uniformly in the extra information needed to add to the state to make the non-Markov process Markov. The technique is illustrated by establishing an inequality in traffic theory concerning the blocking in multi-server service facilities when each customer requires service from several facilities simultaneously.
