The inspection paradox in stochastic resetting

TL;DR

This paper reviews the inspection paradox and explains how stochastic resetting can speed up random processes, using simple mathematical tools to develop intuitive understanding and highlight counter-intuitive effects in probability theory.

Contribution

It provides an accessible explanation of the inspection paradox and its connection to stochastic resetting, offering new insights into how resetting influences process completion times.

Findings

Inspection paradox illustrates counter-intuitive waiting times.

Stochastic resetting can expedite process completion.

Elementary tools help develop probabilistic intuition.

Abstract

The remaining travel time of a plane shortens with every minute that passes from its departure, and a flame diminishes a candle with every second it burns. Such everyday occurrences bias us to think that processes which have already begun will end before those which have just started. Yet, the inspection paradox teaches us that the converse can also happen when randomness is at play. The paradox comes from probability theory, where it is often illustrated by measuring how long passengers wait upon arriving at a bus stop at a random time. Interestingly, such passengers may on average wait longer than the mean time between bus arrivals -- a counter-intuitive result, since one expects to wait less when coming some time after the previous bus departed. In this \textit{viewpoint}, we review the inspection paradox and its origins. The insight gained is then used to explain why, in some situations, stochastic resetting expedites the completion of random processes. Importantly, this is done with elementary mathematical tools which help develop a probabilistic intuition for stochastic resetting and how it works. This viewpoint can thus be used as an accessible introduction to the subject.