  We consider a discrete-time system comprising a first-come-first-served
queue, a non-preemptive server, and a stationary non-work-conserving scheduler.
New tasks enter the queue according to a Bernoulli process with a pre-specified
arrival rate. At each instant, the server is either busy working on a task or
is available. When the server is available, the scheduler either assigns a new
task to the server or allows it to remain available (to rest). In addition to
the aforementioned availability state, we assume that the server has an
integer-valued activity state. The activity state is non-decreasing during work
periods, and is non-increasing otherwise. In a typical application of our
framework, the server performance (understood as task completion probability)
worsens as the activity state increases. In this article, we build on and
transcend recent stabilizability results obtained for the same framework.
Specifically, we establish methods to design scheduling policies that not only
stabilize the queue but also reduce the utilization rate - understood as the
infinite-horizon time-averaged portion of time the server is working. This
article has a main theorem leading to two key results: (i) We put forth a
tractable method to determine, using a finite-dimensional linear program (LP),
the infimum of all utilization rates that can be achieved by scheduling
policies that are stabilizing, for a given arrival rate. (ii) We propose a
design method, also based on finite-dimensional LPs, to obtain stabilizing
scheduling policies that can attain a utilization rate arbitrarily close to the
aforementioned infimum. We also establish structural and distributional
convergence properties, which are used throughout the article, and are
significant in their own right.
