A combinatorial model for lane merging

TL;DR

This paper introduces a combinatorial model for lane merging at intersections, providing formulas for merging paths and connecting to various combinatorial concepts, with proofs based on Andre's Reflection Principle.

Contribution

It develops a novel combinatorial framework for modeling lane merging sequences and derives explicit formulas and connections to classical combinatorial problems.

Findings

Closed formulas for merging paths reaching (n,m) with k zeros

Expected right lane length for sequences with k zeros

Connections to ballot numbers and coin flip sequences

Abstract

A two lane road approaches a stoplight. The left lane merges into the right just past the intersection. Vehicles approach the intersection one at a time, with some drivers always choosing the right lane, while others always choose the shorter lane, giving preference to the right lane to break ties. An arrival sequence of vehicles can be represented as a binary string, where the zeros represent drivers always choosing the right lane, and the ones represent drivers choosing the shorter lane. From each arrival sequence we construct a merging path, which is a lattice path determined by the lane chosen by each car. We give closed formulas for the number of merging paths reaching the point $(n,m)$ with exactly $k$ zeros in the arrival sequence, and the expected length of the right lane for all arrival sequences with exactly $k$ zeros. Proofs involve an adaptation of Andre's Reflection Principle. Other interesting connections also emerge, including to: Ballot numbers, the expected maximum number of heads or tails appearing in a sequence of $n$ coin flips, the largest domino snake that can be made using pieces up to $[n:n]$, and the longest trail on the complete graph $K_n$ with loops.