We are interested in this paper in studying how to schedule tasks in an extreme edge setting, where some sensors produce data and subsequent tasks are executed locally. They are interacting with some external tasks submitted by a superior authority in the cloud. We first model such a system as a problem of scheduling two sets of tasks, each associated with its own objective. The tasks of the first set are released on-line, they can be preempted and the target objective is the minimization of the mean flow-time. The tasks of the second set are known beforehand and cannot be preempted. The objective is to execute them before a common deadline. Due to strong lower bounds on the competitive ratio of this problem, we use the technique of resource augmentation to cope with these limitations. Specifically, our analysis is based on both speed augmentation and rejection. First, we give a general lower bound for the problem, even in the case of speed augmentation and rejection. Then, we propose a competitive algorithm and analyze its performance using dual fitting.