Maximally Stabilizing Task Release Control Policy for a Dynamical Queue

In this paper, we introduce a model of dynamical queue, in which the service time depends on the server utilization history. The proposed queueing model is motivated by widely accepted empirical laws describing human performance as a function of mental arousal. The objective of this paper is to design task release control policies that can stabilize the queue for the maximum possible arrival rate, assuming deterministic arrivals. First, we prove an upper bound on the maximum possible stabilizable arrival rate for any task release control policy. Then, we propose a simple threshold policy that releases a task to the server only if its state is below a certain fixed value. Finally, we prove that this task release control policy ensures stability of the queue for the maximum possible arrival rate.