Modelling user behaviour at a stochastic road traffic bottleneck

Congestion in road traffic has received substantial attention in the research literature. One popular approach to modelling congesting and user response is the seminal bottleneck model introduced by Vickrey [25]. Here traffic is modelled as a fluid, and all travellers are subject to cost for waiting, early departure, and late departure. The travellers' response to the congestion is captured by assuming that they arrive at the bottleneck according to a Wardrop equilibrium, meaning that no traveller can decrease its costs by shifting its arrival time. This model and its extensions have been extensively studied in the research literature, but ignore the fact that road traffic consists of individual travellers with uncertain arrival time and speed. While the fluid approach used in the Vickrey model may be correct when the number of travellers is large, it fails to yield accurate predictions for a small number of travellers.In the present paper we propose a stochastic version of the bottleneck model, that can also handle smaller number of travellers. We discuss the error made by the fluid approximation, and show that the Wardrop equilibrium results in highly varying costs when applied in the more realistic setting with stochasticity. We then discuss an algorithm for numerically computing the equilibrium arrival rate for the stochastic bottleneck model, and propose a closed-form estimation for this equilibrium. This can be used for future studies into the effect of stochasticity in these bottleneck models.