Explaining Freeway Breakdown with Geometric Brownian Motion Model

Abstract

A traffic volume which can trigger a breakdown event at one point in time may not trigger it at another time. The critical question is why a roadway under the same loading is stable in one instance and unstable in another. This paper explains this behavior by using a linear first-order stochastic differential equation (SDE) model, a geometric Brownian motion (gBm) model. This simple stochastic (time-dependent) model of diffusion treats traffic volume, the load on the roadway system, as a random process. The model response variables are (1) the breakdown probability, which is the transition from a free-flow to a congested state for a given traffic loading; and (2) traffic delay. There are two major challenges. The first is formulating an approach, i.e., a mathematical model, that can reliably forecast traffic breakdown at a data collection site located upstream of a bottleneck where no data are collected. The second is selecting and calibrating an appropriate gBm model with extremely volatile data. The approach was assessed by performing match tests, assessing the field data summaries against model forecasts of traffic volume, breakdown probability, and delay. The potential for using the gBm modeling approach as an operational analysis tool was discussed.