Generalized semi-Markov processes (GSMPs) and stochastic Petri nets (SPNs) are generally regarded as performance models (as opposed to logical models) of discrete event systems. Here we take the view that GSMPs and SPNS are essentially automata (generators) driven by input sequences that determine the timing of events. This view combines the deterministic, logical aspects and the stochastic, timed aspects of the two models. We focus on two conditions, (M) and (CX) (which we previously developed to study monotonicity and convexity properties of GSMPs), and the antimatroid and lattice structure they imply for the language generated by a GSMP or SPN. We illustrate applications of these structural properties in the areas of derivative estimation, simulation variance reduction, parallel simulation, and optimal control.