  We introduce and study a non-oriented first passage percolation model having
a property of statistical invariance by time reversal. This model is defined in
a graph having directed edges and the passage times associated with each set of
outgoing edges from a given vertex are distributed according to a generalized
Bernoulli-Exponential law and i.i.d. among vertices. We derive the statistical
invariance property by time reversal through a zero-temperature limit of the
random walk in Dirichlet environment model.
