The totally asymmetric simple exclusion process, which describes the
transport of interacting particles in a lattice, has been actively studied over
the past several decades. For general cases where particles have an extended
size and hop at site-dependent rates, however, theoretically analyzing the
dynamics has remained elusive. Here, we present such an analysis by deriving
and solving the hydrodynamic limit. We obtain closed-form formulas for
steady-state particle densities and currents, as well as phase transition
boundaries. Surprisingly the latter depend on only four parameters: the
particle size and the first, the last, and the minimum hopping rates. Our
results agree well with Monte Carlo simulations and can be used in inference.