Maximally stabilizing task release control policy for a dynamical queue

In this paper, we consider the following stability problem for a novel dynamical queue. Independent and identical tasks arrive for a queue at a deterministic rate. The server spends deterministic state-dependent times to service these tasks, where the server state is governed by its utilization history through a simple dynamical model. Inspired by empirical laws for human performance as a function of mental arousal, we let the service time be related to the server state by a continuous convex function. We consider a task release control architecture which regulates task entry into service. The objective in this paper is to design such task release control policies that can stabilize the dynamical queue for the maximum possible arrival rate, where the queue is said to be stable if the number of tasks awaiting service does not grow unbounded over time. First, we prove an upper bound on the maximum stabilizable arrival rate for any task release control policy by postulating a notion of one-task equilibrium for the dynamical queue and exploiting its optimality. Then, we propose a simple threshold policy that allocates a task to the server only if its state is below a certain fixed value. We prove that this task release control policy ensures stability of the queue for the maximum possible arrival rate.