A computer of a data storage system carries out user input-output requests for data of a set of virtual disks by performing back-end input-output requests to a set of overprovisioned storage devices providing underlying physical storage. A set of time series of utilization measures are continually generated and stored, with each reflecting an amount of a given virtual disk consumed by user data at a given time. The time series are used to generate predicted utilization measures, each being an estimate of an amount of a given virtual disk expected to be consumed by user data at a future time. A fullness probability measure is generated by aggregating the predicted utilization measures, and it reflects a probability that the set of overprovisioned storage devices will be fully consumed by user data at the future time. When the fullness probability measure is above a threshold value, then a system management operation is performed to change operation of the data storage system to avoid full consumption of the overprovisioned storage devices by the virtual disks. For example, the mapping of virtual disks to the overprovisioned storage device is changed in a manner reducing usage of the overprovisioned storage device. The mapping as changed is used for subsequent user input-output requests for data of the virtual disks.