Entropy of sharp restart

Restart has the potential of expediting or impeding the completion times of general random processes. Consequently, the issue of mean-performance takes center stage: quantifying how the application of restart on a process of interest impacts its completion-time’s mean. Going beyond the mean, little is known on how restart affects stochasticity measures of the completion time. This paper is the first in a duo of studies that address this knowledge gap via: a comprehensive analysis that quantifies how sharp restart—a keystone restart protocol—impacts the Shannon entropy of the completion time. The analysis establishes closed-form results for sharp restart with general timers, with fast timers (high-frequency resetting), and with slow timers (low-frequency resetting). These results share a common structure: comparing the completion-time’s hazard rate to a flat benchmark—the constant hazard rate of an exponential distribution whose entropy is equal to the completion-time’s entropy. In addition, using an information-geometric approach based on Kullback-Leibler distances, the analysis establishes results that determine the very existence of timers with which the application of sharp restart decreases or increases the completion-time’s entropy. Our work sheds first light on the intricate interplay between restart and randomness—as gauged by the Shannon entropy.