Maximum Queue Length for Traffic Light with Bernoulli Arrivals

TL;DR

This paper analyzes the maximum queue length at a traffic light with Bernoulli arrivals, providing a distribution for the maximum queue in the case of L=1 and proposing a conjecture for longer red/green cycles.

Contribution

It derives the distribution of maximum queue length for the case L=1 and introduces a conjecture for the case when L>1, advancing understanding of queue behavior in traffic models.

Findings

Derived the distribution of maximum queue length for L=1

Proposed a conjecture for maximum queue distribution when L>1

Analyzed the impact of arrival probability p on queue length

Abstract

Cars arrive at an intersection with a stoplight, which is either red or green. The cars all travel in the same direction, that is, we ignore cross-traffic & oncoming traffic. Assume that the intersection is initially empty. Assume that, at every second, there is a probability p that one new car will arrive at the light, and the outcome is independent of past & future. Let L>=1 be an integer. A red light lasts L seconds; likewise for green. If the light is red, no cars can leave the intersection. If the light is green, cars will leave the intersection at a rate of one per second. Over a time period of n seconds, determine the (random) maximum queue length M of cars at the intersection. What is the distribution of M, as a function of (p,L,n)? We answer this question for the special case L=1 and introduce a conjecture for L>1.