Reliability of weather data processing systems is of prime importance to ensure the efficient operation of space-based weather monitoring systems. This work defines a heterogeneous weather data processing system that is susceptible to uncertainties in data set arrival times. The resource allocation must be robust with respect to these uncertainties. The tasks to be executed by the data processing system are classified into three broad categories: telemetry, tracking and control (high priority); data processing (medium priority); and data research (low priority). The high priority tasks must be completed before considering medium and low priority tasks. The goal of this research is to find a resource allocation that minimizes makespan of the high priority tasks, and to find a mapping that maximizes a function of the completion time and priority of the medium and low priority tasks. Different heuristic techniques to find near optimal solutions are studied, and their performance is evaluated.