Traffic lights, clumping and QBDs

TL;DR

This paper analyzes the distribution of maximum queue lengths at traffic lights with alternating red and green blocks, providing algebraic proofs for certain cases and numerical insights for others.

Contribution

It introduces a novel approach to determine queue length distributions at traffic lights with specific red-green patterns, including algebraic justification and numerical analysis.

Findings

Algebraic asymptotics proven for $oldsymbol{ ext{2 and 3}}$-block cases.

Numerical validation for $oldsymbol{ ext{4 and more}}$-block cases.

Insights into maximum queue length distributions in discrete-time traffic models.

Abstract

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. \ We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics algebraically for $2\leq\ell\leq3$ and numerically for $\ell\geq4$.