Many time-consuming simulators exhibit a complex noise structure that depends on the inputs. Advances in Gaussian process modeling with input-dependent noise, especially via replication (iid repetitions of the same experiment), allow efficient modeling with better uncertainty quantification on the predictions. We focus here on strategies for balancing replication, exploitation and exploration for various sequential design goals, possibly with parallel batches. These goals include global model accuracy, optimization, contour finding, and dimension reduction. Illustration on synthetic examples are provided as well as a large scale massively parallel real world epidemiology problem.