In a distributed heterogeneous computing system, the resources have different capabilities and tasks have different requirements. To maximize the performance of the system, it is essential to assign resources to tasks (match) and order the execution of tasks on each resource (schedule in a manner that exploits the heterogeneity of the resources and tasks. The mapping (defined as matching and scheduling) of tasks onto machines with varied computational capabilities has been shown, in general, to be an NP-complete problem. Therefore, heuristic techniques to find a near-optimal solution to this mapping problem are required. Dynamic mapping is performed when the arrival of tasks is not known a priori. In the heterogeneous environment considered in this study, tasks arrive randomly, tasks are independent (i.e., no communication among tasks), and tasks have priorities and multiple deadlines. This research proposes, evaluates, and compares eight dynamic heuristics. The performance of the best heuristics is 83% of an upper bound.