A Stochastic Differential Equation Framework for Modeling Queue Length Dynamics Inspired by Self-Similarity

Abstract: This article develops a stochastic differential equation (SDE) for modeling the temporal evolution of queue length dynamics at signalized intersections. Inspired by the observed quasiperiodic and self-similar characteristics of the queue length dynamics, the proposed model incorporates three properties into the SDE: (i) mean reversion with periodic mean, (ii) multiplicative noise, and (iii) fractional Brownian motion. It replicates key statistical features observed in real data, including the probability distribution function (PDF) and PSD of queue lengths. To our knowledge, this is the first equation-based model for queue dynamics. The proposed approach offers a transparent, data-consistent framework that may help inform and enhance the design of black-box learning algorithms with underlying traffic physics.