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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.