Stabilizing a Queue Subject to Action-Dependent Server Performance

Understand

We consider a discrete-time system comprising a first-come-first-served queue, a non-preemptive server, and a scheduler that governs the assignment of tasks in the queue to the server.

• The server has an availability state that indicates, at each instant, whether the server is busy working on a task or is available.
• In the latter case, if the queue is nonempty, then a task-assignment control policy implemented by the scheduler either assigns a new task to the server or allows it to rest.
• The server also has an integer-valued activity state that is non-increasing during rest periods, and is non-decreasing otherwise.