Abstract: We present a unified view of likelihood based Gaussian progress regression for simulation experiments exhibiting input-dependent noise. Replication plays a key role in that context as it allows to perform inference for all parameters, bypassing full-data sized calculations. We then borrow a latent-variable idea from machine learning to address heteroskedasticity, leveraging both the computational and statistical efficiency of designs with replication. We further propose to create and sequentially enrich designs with a tunable degree of replication. Illustrations are provided, including real-world simulation experiments from manufacturing and the management of epidemics.