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 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.


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