Probabilistic models for queues at fixed control signals

The estimation of queues at signalized intersections is a classical problem in transportation engineering and operations research. Nevertheless, a general theory able to explain how queues form and cause delays to the drivers is still missing. Typically, queue dynamics are modelled as deterministic, causal phenomena, and under rather limiting assumptions; however, especially in urban networks, these are far from being deterministic or certain. This paper presents a new probabilistic queuing model that can explain the dynamic and stochastic behaviour of queues at fixed controlled signals. The probabilistic approach allows one to capture the temporal behaviour of queues, and to measure the uncertainty of a queue state prediction by computing the evolution of its probability in time, assumed any temporal distribution of the arrivals. This can be fundamental information in, e.g., travel time estimation, network reliability, design and planning of urban areas, and to estimate complex effects that can be observed in congested networks such as spillback or gridlock. Comparison with microscopic simulation shows very good consistency both under the assumption of stationary and non-stationary arrival distributions.