Analytical investigation of oscillations in intersecting flows of   pedestrian and vehicle traffic
  In two intersecting many-particle streams, one can often find the emergence of oscillatory patterns. Here, we investigate the interaction of pedestrians with vehicles, when they try to cross a road. A numerical study of this coupled pedestrian-vehicle delay problem has been presented in a previous paper. Here, we focus on the analytical treatment of the problem, which requires to use a simplified car-following model. Our analytical results for the phase transition to oscillatory pedestrian and traffic flows and the average waiting times are well supported by numerical evaluations and give a detailed picture of the collective dynamics emerging when pedestrians try to cross a road. The mathematical expressions allow one to identify the dependence on model parameters such as the vehicle or pedestrian arrival rate, and the safety factor of pedestrian gap acceptance. We also calculate a formula for the vehicle time gap distribution, which corresponds to the departure time distribution of a M/D/1 queue. 
Dirk Helbing, Rui Jiang, and Martin Treiber