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On Woolhouse's Cotton-Spinning Problem

Jan Friso Groote, Tim A. C. Willemse

TL;DR

The paper revisits Woolhouse's Cotton-Spinning problem by modeling the piecer's walking with probabilistic process formalisms (mCRL2) and quantitative modal logic GW23. It first reproduces Woolhouse's fixed-$N$ single-stroke model and then extends the analysis to multi-stroke scenarios, showing that the original approach tends to overestimate walking distance. A more natural model—where each thread breaks independently with probability $p$—and an optimized variant are introduced, and both yield shorter predicted walking distances, especially for larger mules. Concrete numerical evidence and tooling are provided, including a case with $p=1/220$ and width $46$ giving a relative distance of $0.0761$ (about 8.4 km/day), supporting the conclusion that Woolhouse's estimates were overstated. The work highlights the value of formal probabilistic modeling and offers scalable methods (PRES/PBES, bisimulation) for analyzing industrial-welfare questions in textile production.

Abstract

In 1864 W.S.B. Woolhouse formulated the Cotton-Spinning problem. This problem boils down to the following. A piecer works at a spinning mule and walks back and forth to repair broken threads. The question is how far the piecer is expected to walk when the threads break at random. This problem can neatly be solved using process modelling and quantitative model checking, showing that Woolhouse's model led to an overestimation of the walking distance.

On Woolhouse's Cotton-Spinning Problem

TL;DR

The paper revisits Woolhouse's Cotton-Spinning problem by modeling the piecer's walking with probabilistic process formalisms (mCRL2) and quantitative modal logic GW23. It first reproduces Woolhouse's fixed- single-stroke model and then extends the analysis to multi-stroke scenarios, showing that the original approach tends to overestimate walking distance. A more natural model—where each thread breaks independently with probability —and an optimized variant are introduced, and both yield shorter predicted walking distances, especially for larger mules. Concrete numerical evidence and tooling are provided, including a case with and width giving a relative distance of (about 8.4 km/day), supporting the conclusion that Woolhouse's estimates were overstated. The work highlights the value of formal probabilistic modeling and offers scalable methods (PRES/PBES, bisimulation) for analyzing industrial-welfare questions in textile production.

Abstract

In 1864 W.S.B. Woolhouse formulated the Cotton-Spinning problem. This problem boils down to the following. A piecer works at a spinning mule and walks back and forth to repair broken threads. The question is how far the piecer is expected to walk when the threads break at random. This problem can neatly be solved using process modelling and quantitative model checking, showing that Woolhouse's model led to an overestimation of the walking distance.
Paper Structure (8 sections, 7 equations, 8 figures, 4 tables)

This paper contains 8 sections, 7 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Woolhouse's model
  3. A probabilistic model
  4. Woolhouse's model in mCRL2
  5. A more natural model
  6. A simple natural model
  7. An optimised natural model
  8. Conclusion

Figures (8)

  • Figure 1: Spinning mules in a factory
  • Figure 2: The four walking patterns of the piecer
  • Figure 3: Woolhouse's model in mCRL2
  • Figure 4: The minimal distance covered by the piecer for $\textit{Max}$ repair rounds
  • Figure 5: A more natural model in mCRL2
  • ...and 3 more figures