  The totally asymmetric simple exclusion process (TASEP) is a paradigmatic
stochastic model for non-equilibrium physics, and has been successfully applied
to describe active transport of molecular motors along cytoskeletal filaments.
Building on this simple model, we consider a two-lane lattice-gas model that
couples directed transport (TASEP) to diffusive motion in a semi-closed
geometry, and simultaneously accounts for spontaneous growth and
particle-induced shrinkage of the system's size. This particular extension of
the TASEP is motivated by the question of how active transport and diffusion
might influence length regulation in confined systems. Surprisingly, we find
that the size of our intrinsically stochastic system exhibits robust temporal
patterns over a broad range of growth rates. More specifically, when particle
diffusion is slow relative to the shrinkage dynamics, we observe quasi-periodic
changes in length. We provide an intuitive explanation for the occurrence of
these self-organized temporal patterns, which is based on the imbalance between
the diffusion and shrinkage speed in the confined geometry. Finally, we
formulate an effective theory for the oscillatory regime, which explains the
origin of the oscillations and correctly predicts the dependence of key
quantities, as for instance the oscillation frequency, on the growth rate.
