The dynamics and the uncertainty of queues at fixed and actuated controls: A probabilistic approach

The queue process at controlled junctions plays a fundamental role in many transportation problems, and they are a main determinant of the variability of traffic. Despite the large number of queue models developed in the last 50 years, there is a need for insight into the dynamic and stochastic behavior of traffic at controlled junctions. The existing models are often developed under heuristic, simplifying assumptions, which limit their application validity and interpretation opportunities. This article aims to fill this gap by formulating the queue process as probabilistic. This is done both at fixed and at vehicle actuated controls. Assuming a certain distribution of arrivals and departures one can calculate the probability distribution of queues at fixed control signals. Even if these distributions are assumed stationary, the probability of observing a number of vehicles in queue has a dynamic character. Vehicle actuated controls are in principle designed to adapt the duration of green times according to this dynamic process. By doing that, signal times become dynamic stochastic variables, thus vehicle delays and queues are still random processes in these systems. In this article we compare the performance of these two control types in view of network reliability. The models developed in this research are simple and require small computation times, and they are shown to be consistent with the more elaborate models used in microscopic simulation programs.