Bucket Brigades: Uniqueness of the Fixed Point and Three-Worker Asymptotics

TL;DR

This paper extends the analysis of bucket brigade production lines, focusing on three workers and the uniqueness of fixed points in the no-station setting, revealing new dynamics beyond previous stability regimes.

Contribution

It provides a complete description of three-worker behavior beyond stability and proves fixed point uniqueness for any number of workers in the no-station setting.

Findings

Extended analysis for three workers beyond stable regimes.

Revised understanding of stability conditions.

Proven fixed point uniqueness for arbitrary workers.

Abstract

A standard organization of production lines exhibiting self-balancing behavior is given by bucket brigades. Their study in operations research was initiated by the foundational work of Bartholdi and Eisenstein ({\em Operations Research}, 1996), where a simplified version of the model is considered. Their main result shows that when workers are ordered from the slowest to the fastest, the system is stable and converges to a ``fixed point,'' where each worker oscillates between two limiting positions. They also observe that the dynamics can become highly complex when this ordering condition is not satisfied. The {\em no-station} setting, in which work is distributed continuously and uniformly along the production line, is given special attention in their work. In a subsequent paper with Bunimovich ({\em Operations Research}, 1999), they characterize all stable behaviors of this setting for up to three workers. In this work, we extend their analysis for three workers beyond the stable regime, providing a complete description when workers are ordered from the fastest to the slowest. We also show that, due to their restrictive notion of stability, some of their conclusions must be revisited. Finally, for an arbitrary number of workers, we prove that the fixed point is always unique in the no-station setting.