  We investigate a rich new class of exactly solvable particle systems
generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our
particle systems can be thought of as new exactly solvable examples of tandem
queues, directed first- or last-passage percolation models, or
Robinson-Schensted-Knuth type systems with random input. One of the novel
features of the particle systems is the presence of spatial inhomogeneity which
can lead to the formation of traffic jams.
  For systems with special step-like initial data, we find explicit limit
shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation
results which put the systems into the Kardar-Parisi-Zhang universality class.
At a critical scaling around a traffic jam in the continuous space TASEP, we
observe deformations of the Tracy-Widom distribution and the extended Airy
kernel, revealing the finer structure of this novel type of phase transitions.
  A homogeneous version of a discrete space system we consider is a
one-parameter deformation of the geometric last-passage percolation, and we
obtain extensions of the limit shape parabola and the corresponding asymptotic
fluctuation results.
  The exact solvability and asymptotic behavior results are powered by a new
nontrivial connection to Schur measures and processes.
