  In this paper, we present a stochastic queuing model for the road traffic,
which captures the stationary density-flow relationships in both uncongested
and congestion conditions. The proposed model is based on the $M/g/c/c$ state
dependent queuing model of Jain and Smith, and is inspired from the
deterministic Godunov scheme for the road traffic simulation. We first propose
a reformulation of the $M/g/c/c$ state dependent model that works with
density-flow fundamental diagrams rather than density-speed relationships. We
then extend this model in order to consider upstream traffic demand as well as
downstream traffic supply. Finally, we calculate the speed and travel time
distributions for the $M/g/c/c$ state dependent queuing model and for the
proposed model, and derive stationary performance measures (expected number of
cars, blocking probability, expected travel time, and throughput). A comparison
with results predicted by the $M/g/c/c$ state dependent queuing model shows
that the proposed model correctly represents the dynamics of traffic and gives
good performances measures. The results illustrate the good accuracy of the
proposed model.
