  We consider performing simulation experiments in the presence of covariates.
Here, covariates refer to some input information other than system designs to
the simulation model that can also affect the system performance. To make
decisions, decision makers need to know the covariate values of the problem.
Traditionally in simulation-based decision making, simulation samples are
collected after the covariate values are known; in contrast, as a new
framework, simulation with covariates starts the simulation before the
covariate values are revealed, and collects samples on covariate values that
might appear later. Then, when the covariate values are revealed, the collected
simulation samples are directly used to predict the desired results. This
framework significantly reduces the decision time compared to the traditional
way of simulation. In this paper, we follow this framework and suppose there
are a finite number of system designs. We adopt the metamodel of stochastic
kriging (SK) and use it to predict the system performance of each design and
the best design. The goal is to study how fast the prediction errors diminish
with the number of covariate points sampled. This is a fundamental problem in
simulation with covariates and helps quantify the relationship between the
offline simulation efforts and the online prediction accuracy. Particularly, we
adopt measures of the maximal integrated mean squared error (IMSE) and
integrated probability of false selection (IPFS) for assessing errors of the
system performance and the best design predictions. Then, we establish
convergence rates for the two measures under mild conditions. Last, these
convergence behaviors are illustrated numerically using test examples.
