[Paper Review] Conjectures about Traffic Light Queues

Steven R. Finch, Guy Louchard|arXiv (Cornell University)|Oct 9, 2018

Stochastic processes and statistical mechanics3 references4 citations

TL;DR

This paper proposes conjectured asymptotic distributions for the maximum queue length in a discrete-time traffic light system with periodic red/green cycles of length ℓ ≥ 1. Using numerical solution of large determinantal equations and algebraic recognition from high-precision computations, the authors derive closed-form expressions for the rate constants χₗ(p) in the tail decay of the maximum queue size Mn, extending known results for ℓ=1 to ℓ=2 and ℓ=3. The key contribution is a conjectured extreme value distribution for worst-case congestion under periodic signal control.

ABSTRACT

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. The maximum line length of idle cars is fully understood for $\ell = 1$, but only partially for $2 \leq \ell \leq 3$.

Motivation & Objective

• To understand the worst-case traffic congestion (maximum queue length) in a periodic traffic light system with ℓ-block red and green phases.
• To extend the known exact asymptotic results for ℓ=1 to ℓ≥2, where rigorous treatment is currently infeasible.
• To conjecture closed-form expressions for the rate constants χₗ(p) governing the tail decay of the maximum queue size distribution.
• To develop a computational framework based on high-precision numerical solution of determinantal equations to infer algebraic formulas for χₗ(p).

Proposed method

• Define a stochastic process Sj = max(Sj−1 + Xj, 0) where Xj depends on the phase of the traffic light cycle modulo 2ℓ.
• Model car arrivals as i.i.d. Bernoulli(p) and departures as deterministic -1 during green phases, with no departures during red.
• Construct (k+1)×(k+1) matrices Uk and Vk to represent the transition dynamics over ℓ red and ℓ green phases.
• Numerically solve det[I - Uₖ^ℓ Vₖ^ℓ z] = 0 for the smallest real z > 1 close to unity, yielding zₖ.
• Define χₗ(p) = limₖ→∞ (zₖ - 1)/(p/q)^{2k} to extract the rate constant from the asymptotic behavior.
• Use high-precision numerical values of χₗ(p) for rational p to infer algebraic expressions via polynomial regression and radical recognition.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the maximum queue length Mn in a periodic traffic light system with cycle length 2ℓ and ℓ red/green phases?
• RQ2 How does the rate constant χₗ(p) in the tail decay P(Mn ≤ log(q²/p²)n + h) ∼ exp[−χₗ(p)/(2ℓ)(q²/p²)^{-h}] depend on the arrival probability p and cycle length ℓ?
• RQ3 Can a systematic computational method be developed to conjecture closed-form algebraic expressions for χₗ(p) for ℓ ≥ 2?
• RQ4 How does the worst-case congestion compare across different strategies, including deterministic periodic signals and random signal phases?

Key findings

• For ℓ=2, the conjectured tail decay is P(Mn ≤ log(q²/p²)n + h) ∼ exp[−χ₂(p)/4 ⋅ (q²/p²)^{-h}], with χ₂(p) = (q−p)²/(4q⁶)[(1−8p²+16p³−8p⁴)+(q−p)√(1+4pq)].
• For ℓ=3, the conjectured tail decay is P(Mn ≤ log(q²/p²)n + h) ∼ exp[−χ₃(p)/6 ⋅ (q²/p²)^{-h}], with χ₃(p) expressed as a complex algebraic expression involving radicals and polynomials in p.
• High-precision numerical computation of zₖ from determinantal equations enables recognition of algebraic forms for χₗ(p), verified for p=1/3, 1/5, 1/17, 1/19, yielding exact radical expressions.
• The empirical histograms of maximum queue lengths for n=10¹⁰ match the theoretical predictions extremely well, confirming the conjectured distributions for ℓ=2 and ℓ=3.
• The expected maximum queue length Eₗ(n,p) is minimized for ℓ=1, with ℓ=2 and ℓ=3 performing only slightly worse, while the random signal case (ℓ=0) performs significantly worse.
• The method successfully identifies algebraic structures in χₗ(p), including rational denominators of the form 12(−1+x)^9 when x=1/p is odd, and polynomial coefficients via regression.