On the optimal scheduling of independent, symmetric and time-sensitive tasks

Consider a discrete-time system in which a centralized controller (CC) is tasked with assigning at each time interval (or slot) K resources (or servers) to K out of M\geq K nodes. A node can execute a task when assigned to a server. The tasks are independently generated at each node by stochastically symmetric and memoryless random processes and stored in a finite-capacity task queue. The tasks are time-sensitive since there is a non-zero probability, within each slot, that a task expires before being scheduled. The scheduling problem is tackled with the aim of maximizing the number of tasks completed over time (or the task-throughput) under the assumption that the CC has no direct access to the state of the task queues. The scheduling decisions at the CC are based on the outcomes of previous scheduling commands, and on the known statistical properties of the task generation and expiration processes.