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Lonely passenger problem: the more buses there are, the more lonely passengers there will be

Imre Péter Tóth

TL;DR

The paper analyzes a 'lonely passenger' variant of the balls-into-bins problem: with $n$ passengers choosing uniformly at random among $k$ buses, it shows that increasing the number of buses raises the chance that at least one passenger travels alone and raises the entire distribution of lonely passengers in a stochastic sense. The main technique is a carefully constructed coupling between the $k$- and $k+1$-bus systems, coupled with conditioning on the number of nonempty buses and the use of time-reversed Markov chains to synchronize evolutions. The result is formalized as $L^{(k+1)}_n \succ L^{(k)}_n$ and $p_{n,k+1}>p_{n,k}$, with a rigorous treatment that also connects to Stirling numbers of the second kind. The work highlights subtle probabilistic mechanisms behind monotonicity in occupancy problems and provides a framework for related open problems in probabilistic graph theory.

Abstract

Empty buses are standing at a bus station. $n$ passengers arrive, and they each board a bus completely at random (meaning that they choose uniformly and independently). Then all buses depart. We show that the more buses there are, the more likely it is that someone (i.e. at least one passenger) travels alone (while $n$ is fixed). More generally, we show that the number of lonely passengers increases with the number of buses, in the sense of stochastic dominance. This problem turned out to be surprisingly difficult, with no short solution known to the author so far, despite the efforts of many experts. Some of the results can also be formulated as properties of Stirling numbers of the second kind.

Lonely passenger problem: the more buses there are, the more lonely passengers there will be

TL;DR

The paper analyzes a 'lonely passenger' variant of the balls-into-bins problem: with passengers choosing uniformly at random among buses, it shows that increasing the number of buses raises the chance that at least one passenger travels alone and raises the entire distribution of lonely passengers in a stochastic sense. The main technique is a carefully constructed coupling between the - and -bus systems, coupled with conditioning on the number of nonempty buses and the use of time-reversed Markov chains to synchronize evolutions. The result is formalized as and , with a rigorous treatment that also connects to Stirling numbers of the second kind. The work highlights subtle probabilistic mechanisms behind monotonicity in occupancy problems and provides a framework for related open problems in probabilistic graph theory.

Abstract

Empty buses are standing at a bus station. passengers arrive, and they each board a bus completely at random (meaning that they choose uniformly and independently). Then all buses depart. We show that the more buses there are, the more likely it is that someone (i.e. at least one passenger) travels alone (while is fixed). More generally, we show that the number of lonely passengers increases with the number of buses, in the sense of stochastic dominance. This problem turned out to be surprisingly difficult, with no short solution known to the author so far, despite the efforts of many experts. Some of the results can also be formulated as properties of Stirling numbers of the second kind.
Paper Structure (17 sections, 11 theorems, 40 equations)

This paper contains 17 sections, 11 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Statement of the result
  3. Structure of the proof
  4. Further models and notation
  5. Given number of buses
  6. Strict versus non-strict inequalities
  7. No empty buses
  8. Fixed number of nonempty buses
  9. Preliminaries
  10. Ordering of pure birth processes
  11. Stochastic monotonicity of the number of nonempty buses
  12. Conditional Markov chains
  13. Time reversed Markov chains
  14. Proof of Theorem \ref{['thm:L_k_n_dominance']}
  15. Appendix
  16. ...and 2 more sections

Key Result

theorem 1

For every $n>1$ and $k\ge 1$, $L^{(k+1)}_n \succ L^{(k)}_n$. Also $p_{n,k+1}>p_{n,k}$.

Theorems & Definitions (28)

  • theorem 1
  • lemma 1
  • proof
  • remark 1
  • lemma 2
  • proof
  • remark 2
  • remark 3
  • remark 4
  • lemma 3
  • ...and 18 more