Mean-performance of sharp restart: II. Inequality roadmap

Abstract

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 Lorenz-curve 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.

Diversity of Sharp Restart

“…In turn, various statistics of the output can be further calculated, e.g. : the output's mean (which was the focus of the studies [18]- [19]), as well as higher-order moments of the output; the output's Boltzmann-Gibbs-Shannon entropy (which was the focus of the first part of this duo [1]); and the output's diversity (which is the focus of this paper). So, why bother with the mean-performance analyses of [18]- [19], and with the randomness analyses of this duo?…”

“…As established in the mean-performance analyses of [18]- [19] and in the entropy analysis of [1], and as shall be established here: 'standing on the shoulders' of Eq. ( 1), it is possible to construct highly insightful and highly implementable results that altogether circumvent the need to use Eq.…”

“…Applying sharp restart, the 'input' task-completion time T is mapped to the 'output' taskcompletion time T R . The studies [18]- [19] examined the effect of sharp restart on mean-performance: is the output's mean smaller or larger than the input's mean? In other words, does the mean task-completion time decrease…”

Diversity of sharp restart

“…Among general restart protocols, sharp restart assumes center stage due to the following key fact: from a mean-performance perspective, sharp restart can out-perform any other restart protocol [2,3]. A comprehensive mean-performance analysis of sharp restart was established and presented in [18,19].…”

“…In essence, a similar situation holds with regard to sharp restart. Indeed, equipped only with a mean-performance analysis of sharp restart [18,19], we are completely 'in the dark' with regard to the effect of sharp restart on randomness. Namely, we do not know if the application of sharp restart decreases or increases the inherent randomness of task-completion durations.…”