The slow-to-start mechanism is known to play an important role in the particular shape of the Fundamental diagram of traffic and to be associated to hysteresis effects of traffic flow. We study this question in the context of stochastic processes, namely exclusion and queueing processes, by including explicitly an asymmetry between deceleration and acceleration in their formulation. Spatial condensation phenomena and metastability is observed, depending on the level of the aforementioned asymmetry. The relationship between these 2 families of odels is analyzed on the ring geometry, to yield a large deviation formulation of the fundamental diagram (FD)