A Note on Walking Versus Waiting

TL;DR

This paper analyzes optimal strategies for travelers at bus stops to decide when to wait or walk, considering various probability distributions of bus arrivals and intermediate stops, revealing nuanced decision rules.

Contribution

It provides a comprehensive mathematical analysis of waiting versus walking strategies under general and specific probabilistic models, including the uniform headway case.

Findings

Optimal strategies depend on headway length and confidence levels.

Waiting is optimal if headway is less than walking time.

Walking is preferable if headway exceeds twice the walking time.

Abstract

This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and solution is provided for a very general class of probability distributions of bus arrival time, and the solution characterised in terms of a function analogous to the hazard rate in reliability theory. The note also considers the question of intermediate stops. It is found that the optimal strategy is not always the laziest, even when headways are not excessively long. For the common special case where one knows the (uniform) headway but not the exact timetable, it is shown that one should wait if the headway is less than the walking time (less bus travel time), and walk if the headway is more than twice this much. In between it may be better to wait or to walk, depending on one's confidence in being able to catch up to a passing bus.