The Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem

TL;DR

This paper uses the Michaelis-Menten reaction scheme to analyze and solve the optimal restart problem, providing a unified framework that reveals universal scaling laws and has applications in stochastic processes and algorithms.

Contribution

It introduces a novel approach applying Michaelis-Menten kinetics to the optimal restart problem, deriving solvable equations and uncovering universal regimes and scaling laws.

Findings

Exact solutions for specific restart regimes

Universal, details-independent solution forms

Optimal scaling laws for restart rates

Abstract

We study the effect of restart, and retry, on the mean completion time of a generic process. The need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of Michaelis and Menten. Stopping a process in its midst, only to start it all over again, may prolong, leave unchanged, or even shorten the time taken for its completion. Here we are interested in the optimal restart problem, i.e., in finding a restart rate which brings the mean completion time of a process to a minimum. We derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest. We then continue to discover regimes at which solutions to the problem take on universal, details independent, forms which further give rise to optimal scaling laws. The formalism we develop, and the results obtained, can be utilized when optimizing stochastic search processes and randomized computer algorithms. An immediate connection with kinetic proofreading is also noted and discussed.