I will describe recent advances in bringing a circle of ideas and techniques around Bethe ansatz and Yanga**Baxter relation under the probabilistic roof, which provides new examples of stochastic interacting particle systems, and techniques to solve them.

In particular,I plan to discuss a new particle dynamics in continuous inhomogeneous medium with features resembling traffic models, as well as queuing systems. This system has phase transitions (discontinuities in the limit shape) and Tracy-Widom fluctuations (even at the point of the phase transition).