In many applications of wireless sensor actor networks (WSANs) that often run in harsh environments, the reduction of completion times of tasks is highly desired. We present a new time-aware, energy-aware, and starvation-free algorithm called Scate for assigning tasks to actors while satisfying the scalability and distribution requirements of WSANs with semi-automated architecture. The proposed algorithm allows concurrent executions of any mix of small and large tasks and yet prevents probable starvation of tasks. To achieve this, it estimates the completion times of tasks on each available actor and then takes the remaining energies and the current workloads of these actors into account during task assignment to actors. The results of our experiments with a prototyped implementation of Scate show longer network lifetime, shorter makespan of resulting schedules, and more balanced loads on actors compared to when one of the three well-known task-scheduling algorithms, namely, the max-min, min-min, and opportunistic load balancing algorithms, is used.