We present a fully Bayesian sequential strategy for predicting the mean response surface of heteroscedastic stochastic simulation models. Leveraging dual Gaussian processes as the surrogate and a selection criterion based on expected Bayesian integrated mean-square prediction error, our approach sequentially selects informative design points while fully accounting for parameter uncertainty. Sequential importance sampling is employed to efficiently update the posterior distribution of the parameters. Our strategy is tailored for expensive simulation models, where achieving robust predictive accuracy under a limited budget is critical. Using synthetic examples, we illustrate its practical advantages compared to existing approaches, in terms of predictive accuracy, noise estimation, and uncertainty quantification. We then implement the proposed strategy on a real motivating application in seismic design of wood-frame podium buildings.

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