A Queueing-Theoretic Foundation for Optimal Spaced Repetition

In the study of human learning, there is broad evidence that our ability to retain a piece of information improves with repeated exposure, and that it decays with delay since the last exposure. This plays a crucial role in the design of educational software, leading to a trade-off between teaching new material and reviewing what has already been taught. A common way to balance this trade-off is spaced repetition, which uses periodic review of content to improve long-term retention. Though spaced repetition is widely used in practice, e.g., in electronic flashcard software, there is little formal understanding of the design of these systems. Our paper addresses this gap in three ways. First, we mine log data from spaced repetition software to establish the functional dependence of retention on reinforcement and delay. Second, we use this memory model to develop a stochastic model for spaced repetition systems. In particular, we propose a queueing-network model of the Leitner system for reviewing flashcards, and empirically test the validity of our queueing model via a Mechanical Turk experiment. Our experiments verify a key qualitative prediction of our model: the existence of a sharp phase transition in learning outcomes upon increasing the rate of new item introductions. Finally, we propose a heuristic approximation for our queueing model that provides a computationally tractable means of optimizing the design of the Leitner system. We discuss various aspects of the resulting optimal design, and also outline ways of extending our model to study other aspects of designing spaced repetition systems.