  We consider traffic flows described by conservation laws. We study a 2:1
junction (with two incoming roads and one outgoing road) or a 1:2 junction
(with one incoming road and two outgoing roads). At the mesoscopic level, the
priority law at the junction is given by traffic lights, which are periodic in
time and the traffic can also be slowed down by periodic in time flux-limiters.
After a long time, and at large scale in space, we intuitively expect an
effective junction condition to emerge. Precisely, we perform a rescaling in
space and time, to pass from the mesoscopic scale to the macroscopic one. At
the limit of the rescaling, we show rigorous homogenization of the problem and
identify the effective junction condition, which belongs to a general class of
germs (in the terminology of [6, 21, 37]). The identification of this germ and
of a characteristic subgerm which determines the whole germ, is the first key
result of the paper. The second key result of the paper is the construction of
a family of correctors whose values at infinity are related to each element of
the characteristic subgerm. This construction is indeed explicit at the level
of some mixed Hamilton-Jacobi equations for concave Hamiltonians (i.e. fluxes).
The explicit solutions are found in the spirit of representation formulas for
optimal control problems.
