Formal methods for task management for human operators are gathering increasing attention to improve efficiency of human-in-the-loop systems. In this paper, we consider a dynamical queue approach to task management for human operators. We consider the model of dynamical queue proposed in our earlier work [1], in which the service time depends on the server utilization history. The focus of the paper is to characterize the throughput of the dynamical queue and design corresponding maximally stabilizing task release control policies, assuming deterministic arrivals. We focus extensively on threshold policies that release a task to the server only when the server state is less than a certain threshold. When every task brings in the same deterministic amount of work, we give an exact characterization of the throughput and show that an appropriate threshold policy is maximally stabilizing. When the amount of work associated with the tasks is an i.i.d. random variable with finite support, we show that the maximum throughput increases in comparison to the case where the tasks have deterministic amount of work.
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