Mean-performance of sharp restart: II. Inequality roadmap

Restarting a deterministic process always impedes its completion. However, it is known that restarting a random process can lead to an opposite outcome-expediting completion. Hence, the effect of restart is contingent on the underlying statistical heterogeneity of the process' completion times. To quantify this heterogeneity we introduce a novel approach to restart research: the methodology of inequality indices, which is widely applied in economics and in the social sciences to measure income and wealth disparities. Utilizing this approach we establish an 'inequality roadmap' for the mean-performance of sharp restart: a whole new set of universal inequality criteria that determine when restart with sharp timers (i.e. with fixed deterministic timers) impedes/expedites mean completion. The criteria are based on key Lorenzcurve inequality indices including Bonferroni, Gini, and Pietra. From a practical perspective, the criteria offer researchers highly useful tools to tackle the common real-world situation in which only partial information of the completion-time statistics is available. From a theoretical perspective, the criteria yield-with unprecedented precision and resolution-a powerful and overarching take-home-message: restart impedes/expedites mean completion when the underlying statistical heterogeneity is low/high, respectively.As sharp restart can match the mean-performance of any other restart protocol, the results established here apply to restart research at large.