  We study asymmetric exclusion processes (TASEP) on a nonuniform
one-dimensional ring consisting of two segments having unequal hopping rates,
or {\em defects}. We allow weak particle nonconservation via Langmuir kinetics
(LK), that are parameterised by generic unequal attachment and detachment
rates. For an extended defect, in the thermodynamic limit the system
generically displays inhomogeneous density profiles in the steady state - the
faster segment is either in a phase with spatially varying density having no
density discontinuity, or a phase with a discontinuous density changes.
Nonequilibrium phase transitions between them are controlled by the
inhomogeneity and LK. The slower segment displays only macroscopically uniform
bulk density profiles in the steady states, reminiscent of the maximal current
phase of TASEP but with a bulk density generally different from half. With a
point defect, there are low and high density spatially uniform phases as well,
in addition to the inhomogeneous density profiles observed for an extended
defect. In all the cases, it is argued that the the mean particle density in
the steady state is controlled only by the ratio of the LK attachment and
detachment rates.
