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 with 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. History: Accepted by Bruno Tuffin, area editor for simulation. Funding: This work was supported in part by Singapore Ministry of Education Academic Research Funds [Tier 1 Grants R-155-000-201-114 and A-0004822-00-00], the City University of Hong Kong [Grants 7005269 and 7005568], and the National Natural Science Foundation of China [Grant 72091211]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1263 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0329 ) at ( http://dx.doi.org/10.5281/zenodo.7344997 ).

Consider a set of n tasks with a tree-structured precedence relation and execution time of 1 or 3 units. We give an $O(n^2 \log n)$-time algorithm to find a minimum length schedule for these tasks on two identical processors. Possible generalization to the case of 1 or k units is also given.

One of the assumptions made in classical scheduling theory is that a task is always executed by one processor at a time. With the advances in parallel algorithms, this assumption may not be valid for future task systems. In this paper, a new model of task systems is studied, the so-called Parallel Task System, in which a task can be executed by one or more processors at the same time. The complexity of scheduling Parallel Task Systems to minimize the schedule length is examined. For nonpreemptive scheduling, it is shown that the problem is strongly NP-hard even for two processors when the precedence constraints consist of a set of chains. For independent tasks, the problem is strongly NP-hard for five processors, but solvable in pseudo-polynomial time for two and three processors. For preemptive scheduling, it is shown that the problem is strongly NP-hard for arbitrary number of processors for a set of independent tasks. Furthermore, it is shown that it is NP-hard, but solvable in pseudo-polynomial time, for a fixed number of processors.

The problem of preemptively scheduling a task system consisting of a set of n independent tasks on one processor so as to minimize the mean flow time is considered. The goal is to find a preemptive schedule such that the mean flow time is minimized subject to the constraint that task T/sub i/ is executed within the interval between its release time and its deadline. Such a schedule, if it exists, is called an optimal schedule. It is shown that the problem of finding an optimal schedule is NP-hard. A greedy algorithm is given to find an optimal schedule for a large class of task systems.>