  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.
