Queueing Subject to Activity-Dependent Server Performance: Task-Assignment Control Policies for Utilization Rate Reduction - Lin, Michael; Martins, Nuno C.; La, Richard J. (IEEE transactions on control of network systems, 2022)
We consider a discrete-time system comprising a first-come-first-served queue, a nonpreemptive server, and a scheduler that assigns tasks from the queue to the server. New tasks enter the queue according to a Bernoulli process with a prespecified arrival rate. At each instant, the server is either working on a task or is available. The scheduler implements a task-assignment control policy that we seek to design. When the server is available and the queue is nonempty, the policy 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 nondecreasing during work periods, and is nonincreasing 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 task-assignment control policies, which we call scheduler policies for short, which 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: 1) 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 scheduler policies that are stabilizing, for a given arrival rate. 2) We propose a design method, also based on finite-dimensional LPs, to obtain stabilizing scheduler 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 this article, and are significant in their own right.